symmetric algebra

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## Idea

The **symmetric algebra** $S V$ of a vector space is the [[free object|free]] [[commutative algebra]] over $V$.

This construction generalizes to [[group representations]], [[chain complexes]], [[vector bundles]], [[coherent sheaves]], and indeed objects in any [[symmetric monoidal category|symmetric monoidal]]  [[linear category|linear categories]] with enough [[colimit|colimits]], where the tensor product distributes over those colimits (as in a [[2-rig]]).

## Explicit definition

We begin with the construction for vector spaces and then sketch how to generalize it.

### For vector spaces

Suppose $V$ is a [[vector space]] over a [[field]] $K$.  Then the **symmetric algebra** $S V$ is generated by the elements of $V$ using these operations:

*  addition and scalar multiplication
*  an associative binary operation $\cdot$

subject to these identities:

*  the identities necessary for $S V$ to be an [[associative algebra]]
*  the identity $v \cdot w = w \cdot v$ for all $v \in V$.

It then follows that $S V$ is a [[graded algebra]]
where $S^p V$ is spanned by $p$-fold products, that is, elements of the form

$$v_1 \cdot \cdots \cdot v_p $$

where $v_1, \dots, v_p \in V$.  Clearly $S V$ is also [[commutative algebra|commutative]].

The symmetric algebra of $V$ is also denoted $Sym V$.   It is also called the **polynomial algebra**.  However we should carefully distinguish between polynomials _in_ the elements of $V$, which form the algebra $S V$, and polynomial functions _on_ the vector space $V$, which form the algebra $S(V^*)$.    In quantum physics, a similar construction for Hilbert spaces is known as the [[Fock space]].

### In general

More generally, suppose $C$ is any [[symmetric monoidal category]] and $V \in C$ is any object.  Then we can form the [[tensor power|tensor powers]] $V^{\otimes n}$.  If $C$ has countable coproducts we can form the [[coproduct]]

$$  T V = \bigoplus_{n \ge 0} V^{\otimes n} $$

(which we write here as a [[direct sum]]), and if the tensor product distributes over these coproducts, $T V$ becomes a [[monoid]] object in $C$, with multiplication given by the obvious maps

$$   V^{\otimes p} \otimes V^{\otimes q} \to V^{\otimes (p+q)} $$

This monoid object is called the [[tensor algebra]] of $V$.

The [[symmetric group]] $S_n$ acts on $V^{\otimes n}$, and if $C$ is a [[linear category]] over a [[field]] of [[characteristic]] zero, then we can form the symmetrization map

$$  p_A : V^{\otimes n} \to V^{\otimes n} $$

given by

$$ p_A = \frac{1}{n!} \sum_{\sigma \in S_n} \sigma $$

This is an [[idempotent]], so if idempotents split in $C$ we can form its [[image]], called the $n$th **symmetric tensor power** or **symmetric power** $S^n V$ (e.g [Gallier 2011, p. 27](#Gallier11)). The coproduct

$$  S V = \bigoplus_{n \ge 0} S^n V $$

becomes a monoid object called the **symmetric algebra** of $V$.

If $C$ is a more general sort of symmetric monoidal category, then we need a different construction of $S^n V$.  For example, if $C$ is a symmetric monoidal category with finite colimits, we can simply define $S^n V$ to be the [[coequalizer]] of the action of the symmetric group $S_n$ on $V^{\otimes n}$.  And if $C$ also has countable coproducts, we can define

$$  S V = \coprod_{n \ge 0} S^n V $$

Then, if the tensor product distributes over these colimits (as in a [[2-rig]]), $S V$ will become a [[internalization|commutative monoid object]] in $C$.  In fact, it will be the _free_ commutative monoid object on $V$, meaning that any morphism in $C$

$$ V \to A \, , $$

where $A$ is a commutative monoid, factors uniquely as the obvious morphism

$$ V \to S V $$

followed by a morphism of commutative monoids

$$ S V \to A  \, ,$$

as in this commutative triangle:

$$ \array {
     &          & S V \\
     & \nearrow &                 & \searrow \\
   V &          & \longrightarrow &          & A
} $$

### In even more generality

If $V$ is an object in a [[CMon-enriched symmetric monoidal category]] $\mathcal{C}$ such that the following diagram admits a joint [[coequalizer]] for every $n \ge 2$:

\begin{tikzcd}
V^{\otimes n} \arrow[rr, "\sigma", shift left=2] \arrow[rr, "...", shift right=2] &  & V^{\otimes n}
\end{tikzcd}

(there is one morphism for every $\sigma \in \mathfrak{S}_{n}$, such permutation natural transformations being defined in the entry [[symmetric monoidal category]]),

we say that $V$ possess all the symmetric powers. The $n^{th}$ symmetric power is given by the coequalizer:

\begin{tikzcd}
V^{\otimes n} \arrow[rr, "r_{n}"] &  & S_{n}(V)
\end{tikzcd}

We put $S_{0}(V)=I$ and $S_{1}(V)=V$.

If the category possess the countable coproducts, then we can form

$$S(V) = \underset{n \ge 0}{\coprod}S_{n}(V)$$

As above, if the tensor product distributes over the countable coproducts and preserve the finite colimits in each argument, then $S(V)$ will be the free [[commutative monoid object]] on V.

For example, in the category [[Rel]] of sets and relations, this construction gives the set $\mathcal{M}(V)$ of multisets with elements in the set $V$.

## Examples

+-- {: .num_example #SymmetricAlgebraInCoChainComplexes}
###### Example
**(symmetric algebra in [[chain complexes]] is [[differential graded-commutative algebra]])**

Let the ambient [[category]] be the [[category of cochain complexes]] over a [[ground field]] of [[characteristic zero]], regarded as a [[symmetric monoidal category]] via the [[tensor product of chain complexes]].

Then for $(V^\bullet, d)$ a [[cochain complex]], the symmetric algebra

$$
  Sym\left( (V^\bullet, d) \right)
  \;\in\; CMon( Ch^\bullet, \otimes )
$$

is the [[differential graded-commutative algebra]] whose underlying [[graded algebra]] is the [[graded-commutative algebra]] on $V^\bullet$, and whose [[differential]] is the original $d$, extended, uniquely, as a graded [[derivation]] of degree +1.

=--

## Properties

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

Let the ambient [[category]] be the [[category of cochain complexes]] over a [[ground field]] of [[characteristic zero]], regarded as a [[symmetric monoidal category]] via the [[tensor product of chain complexes]] and consider the [[differential graded-commutative algebra]] $Sym(V^\bullet,d)$ free on a cochain complex $(V^\bullet,d)$ from example \ref{SymmetricAlgebraInCoChainComplexes}.

Then the [[cochain cohomology]] (of the [[forgetful functor|underlying]] cochain complex) of $Sym(V^\bullet,d)$ is the graded symmetric algebra on the [[cochain cohomology]] of $(V^\bullet,d)$:

$$
  H^\bullet\left(
    Sym\left(V^\bullet,d\right)
  \right)
  \;\simeq\;
  Sym\left( H^\bullet(V^\bullet,d) \right)
  \,.
$$

=--

See also [this MO discussion](https://mathoverflow.net/a/91243/381).

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

We have the following sequence of [[linear isomorphisms]]:

$$
  \begin{aligned}
    H^\bullet
    \left(
      Sym( V^\bullet,d )
    \right)
    & \simeq
    H^\bullet
    \left(
      \underset{k \in \mathbb{N}}{\oplus}
      \left(
        (V^\bullet,d)^{\otimes_k}
      \right)^{\Sigma_k}
    \right)
    \\
    & \simeq
    \underset{k \in \mathbb{N}}{\oplus}
    H^\bullet\left(
       \left( (V^\bullet,d)^{\otimes_k}\right)^{\Sigma_k}
    \right)
    \\
    & \simeq
    \underset{k \in \mathbb{N}}{\oplus}
    H^\bullet\left(
       \left( (V^\bullet,d)^{\otimes_k}\right)
    \right)^{\Sigma_k}
    \\
    & \simeq
    \underset{k \in \mathbb{N}}{\oplus}
    \left(
      H^\bullet\left(
        V^\bullet,d
      \right)^{\otimes_k}
    \right)^{\Sigma_k}
    \\
    & =
    Sym\left( H^\bullet(V^\bullet,d)\right)
  \end{aligned}
$$

Here:

1. the first step uses that, while _a priori_ the symmetric algebra is equivalently the [[quotient]] of the [[tensor algebra]] by the [[symmetric group]] [[action]], in [[characteristic zero]] this is equivalently [[invariants]] of the [[symmetric group]] [[action]], because here $V^G \to V \to V_G$ is a [[linear isomorphism]];

1. the second step uses that [[cochain cohomology]] respects [[direct sums]];

1. the third step uses that for [[finite groups]] in [[characteristic zero]], taking [[invariants]] is compatible with passing to [[cochain cohomology]] ([this prop.](invariant#TakingInvariantsForFiniteGroupCommutesWithTaingHomologyInCharZero)):

1. the fourth step is the [[Künneth theorem]] for [[ordinary homology]] over a [[field]] ([this prop.](Künneth+theorem#InordinaryHomology)).

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Symmetric powers in a symmetric monoidal $\mathbb{Q}^{+}$-linear category  are characterized among the countable families of objects as forming a special connected graded quasi-bialgebra (reference to come).

## Related concepts

* [[symmetric algebra spectrum]]

* [[symmetric ring]]

* [[symmetric power]]

## References

* [[Bourbaki]], _Algèbre_, chap. III, § 6.

* [[Stacks Project]], [Tag 00DM](http://stacks.math.columbia.edu/tag/00DM).

* {#Gallier11} [[Jean Gallier]], *Tensor Algebras, Symmetric Algebras and Exterior Algebras* [[pdf](https://www.cis.upenn.edu/~cis6100/diffgeom7.pdf)], section 22 in: *Notes on Differential Geometry and Lie Groups* (2011)

* {#Gallier20} [[Jean Gallier]], [[Jocelyn Quaintance]], *Tensor Algebras and Symmetric Algebras*, Ch 2 in: *Differential Geometry and Lie Groups -- A second course*, Geometry and Computing **13**, Springer (2020) [[doi:10.1007/978-3-030-46047-1](https://doi.org/10.1007/978-3-030-46047-1), [webpage](https://www.cis.upenn.edu/~jean/gbooks/manif.html), [pdf](https://www.seas.upenn.edu/~jean/diffgeom-spr-II.pdf)]

On the [[symmetric algebra spectrum]] of the [[sphere spectrum]], and its structure as a [[Hopf ring spectrum]]:

* [[Neil Strickland]], [[Paul Turner]], _Rational Morava $E$-theory and $D S^0$_, Topology **36** 1 (1997) 137-151 ([pdf](http://hopf.math.purdue.edu/Strickland-PTurner/rme.pdf))

Further on symmetric powers in [[homotopical algebra|homotopical]]/[[higher algebra]]:

* {#GorchinskiyGuletskii16} [[Sergey Gorchinskiy]], [[Vladimir Guletskii]], *Symmetric powers in abstract homotopy categories*, Adv. Math., **292** (2016) 707-754 [[arXiv:0907.0730](https://arxiv.org/abs/0907.0730), [doi:10.1016/j.aim.2016.01.011](https://doi.org/10.1016/j.aim.2016.01.011)]

Discussion via [[string diagrams]]:

* {#Vienney23} [[Jean-Baptiste Vienney]], *String diagrams for symmetric powers I: In symmetric monoidal $\mathbb{Q}_{\geq 0}$-linear categories* [[arXiv:2308.02094](https://arxiv.org/abs/2308.02094)]

[[!redirects symmetric algebras]]
[[!redirects symmetric power]]
[[!redirects symmetric powers]]
[[!redirects symmetric tensor power]]
[[!redirects symmetric tensor powers]]

[[!redirects symmetric product]]
[[!redirects symmetric products]]

[[!redirects symmetric tensor algebra]]
[[!redirects symmetric tensor algebras]]


symmetric bicategory

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## Definition

\begin{definition}
A **symmetric bicategory** ([May & Sigurdsson 2006, Def. 16.2.1](#MaySigurdsson06)) is a [[bicategory]] $B$ equipped with a [[bijective-on-objects]] [[equivalence of 2-categories|biequivalence]] $t \colon B \simeq B^{op}$ to its [[opposite 2-category]].

A **closed symmetric bicategory** is a symmetric bicategory that is [[closed bicategory|closed]] ([May & Sigurdsson 2006, Def. 16.3.1](#MaySigurdsson06)).
\end{definition}

\begin{remark}
The notion of symmetric bicategory is a [[horizontal categorification]] of that of a  [[symmetric monoidal category]]. It is not to be confused with the notion of a  symmetric [[monoidal bicategory]], which is instead a [[vertical categorification]] of the same concept.

Note, however, that single-object symmetric bicategories are more expressive than [[symmetric monoidal categories]]: a symmetric bicategory with a single object, $Obj(B) \simeq \{\ast\}$, is a symmetric monoidal category if and only if the component functor of $t$ is the [[identity functor]] on the single [[hom-category]] $B(\ast,\ast)$.
\end{remark}

A symmetric bicategory is a [[categorified]] [[dagger-category]].

## Examples

\begin{example}
[[Span]] is a symmetric bicategory, whose involution is given by reversing each span.
\end{example}

\begin{example}
[[Prof]] is a closed symmetric bicategory, whose involution is induced by the [[duality involution]] on [[Cat]].
\end{example}

## Related concepts

* [[closed bicategory]]

* [[duality involution]]

* [[dagger category]]

## References

The notion of symmetric bicategories was introduced in Definition 16.2.1 (of the published [[MaySigurdsson_ParametrizedHomotopyTheory.pdf:file]]-version, not in the arXiv version, of):

* {#MaySigurdsson06} [[Peter May]], [[Johann Sigurdsson]], _[[Parametrized Homotopy Theory]]_,  Mathematical Surveys and Monographs, vol. 132, AMS 2006   ([ISBN:978-0-8218-3922-5](https://bookstore.ams.org/surv-132), [pdf](http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf), [[MaySigurdsson_ParametrizedHomotopyTheory.pdf:file]], [arXiv:math/0411656](https://arxiv.org/abs/math/0411656))

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[[!redirects closed symmetric bicategory]]
[[!redirects closed symmetric bicategories]]
[[!redirects symmetric closed bicategory]]
[[!redirects symmetric closed bicategories]]


symmetric cancellative midpoint algebra

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## Idea

A [[symmetric midpoint algebra]] that is also a [[cancellative midpoint algebra]].

## Definition

A __symmetric cancellative midpoint algebra__ is a [[symmetric midpoint algebra]] $(M,\vert, \odot, (-)^\bullet)$ that satisfies the cancellative property:

* for all $a$, $b$, and $c$ in $M$, if $a \vert b = a \vert c$, then $b = c$

## Properties

For all $a$ and $b$ in $M$, $a = b$ if and only if $a^\bullet \vert b = \odot$.

## Examples

The [[rational numbers]], [[real numbers]], and the [[complex numbers]] with $a \vert b \coloneqq \frac{a + b}{2}$, $\odot = 0$, and $a^{\bullet} = -a$ are examples of symmetric cancellative midpoint algebras.

The [[trivial group]] with $a \vert b = a \cdot b$, $\odot = 1$ and $a^{\bullet} = a^{-1}$ is a symmetric cancellative midpoint algebra.

## Related concepts

* [[cancellative midpoint algebra]]

* [[symmetric 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))


symmetric closed midpoint algebra

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## Idea

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

## Definition

A __symmetric closed midpoint algebra__ is a [[symmetric cancellative midpoint algebra]] $(M,\vert, \odot, (-)^\bullet)$ with two elements $\bot:M$ and $\top:M$ such that $\bot\vert\top = \odot$.

## Properties

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

## Examples

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

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

## Related concepts

* [[closed midpoint algebra]]

* [[symmetric cancellative midpoint algebra]]

* [[minor scale]]

* [[dyadic rational module]]

## 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))


symmetric comonad

Given a [[comonad]] $\mathbf{G}=(G,\delta,\epsilon)$ on a category $A$, a [[natural transformation]]
$t : G G\rightarrow G G$ is a **symmetry** of $\mathbf{G}$
if it sastisfies the quantum [[Yang-Baxter equation]] (transposed braid relation)

$$
G(t) t_G G(t) = t_G G(t) t_G
$$

and also

\begin{equation}\label{symmG}
t^2 = 1_{G G},\,\,\,\,t\delta = \delta,\,\,
\,\,\,\epsilon_G t = G(\epsilon),\,\,\,
\delta_G t = G(t) t_G G(\delta)\end{equation}
In that case, the pair $(G,t)$ is called a **symmetric comonad**.

The [[simplicial set]] obtained by the standard construction from a symmetric comonad has always a structure of a [[symmetric simplicial set]].

* M. Grandis, Finite sets and symmetric simplicial sets,
Theory of Applications of Categories,
Vol. 8, No. 8, pp. 244--253, [link]() (to come?).

* Z. Škoda, Cyclic structures for simplicial objects from comonads, [math.CT/0412001](http://arxiv.org/abs/math.CT/0412001).


symmetric difference

The __symmetric difference__ of two sets $A$ and $B$ (either [[pure sets]] or [[subsets]] of some fixed [[set]]), written $A \uplus B$ (and a host of other ways), may be defined using [[exclusive disjunction]]:
$$ A \uplus B = \{ x \;|\; x \in A \;⊻\; x \in B \} .$$
You can also call this [[exclusive union]], especially if you want to use it analogously to exclusive disjunction; then the exclusive union of $n$ sets consists of those elements $x$ that belong to exactly one of the $n$ sets.

The term _symmetric difference_, however, is usually used as the addition in a [[Boolean ring]] and so interpreted as an associative operation; then the symmetric difference of $n$ sets consists of those elements $x$ that belong to an *odd* number of the $n$ sets.

The [[empty set]] is the [[identity element|identity]] for this operation; it is the symmetric difference (or exclusive union, for that matter) of no sets.

In [[constructive mathematics]], the symmetic difference cannot be proved associative and so becomes less useful (unless one restricts attention to sets with [[decidable equality]], where it can be proved associative).  This is a problem in [[measure theory]]; see [[Cheng measurable space]] for one way to fix this.

Note that the [[union]], symmetric difference, and (internal, or external up to [[natural isomorphism]]) [[disjoint union]] of a family of (pairwise) [[disjoint sets]] are all the same.  For a family that is not disjoint, however, the union and symmetric difference are different, the internal disjoint union does not make sense, and the external disjoint union is not isomorphic to either the union or the symmetric difference (at least not naturally, and in some cases not at all).


symmetric endofunctor

## Idea

A _symmetric endofunctor_ is an [[endofunctor]] together with [[actions]] of the [[symmetric groups]] on its [[automorphism groups]].

## Definition

Let $\Phi$ be a [[graded monoid]] in the [[category]] of [[groups]], e.g. the [[graded monoid]] $\Sigma = (\Sigma_n)_{n \ge 0}$ of [[symmetric groups]].

A **symmetric endofunctor** $F : C \to C$ is an [[endofunctor]] together with the data of, for each $n \in \mathbf{N}$, a [[group homomorphism]]
  $$ \Phi_n \to \Aut(F^{\circ n}) $$
where $F^{\circ n}$ is the $n$-fold [[composite]] of $F$.


symmetric function

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# Symmetric functions
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## Idea

A _symmetric function_ is roughly a [[polynomial]] that is invariant under [[permutation]] of its [[variables]].  However, this is only strictly correct if the number of variables is [[finite number|finite]], while symmetric functions depend on a countably infinite number of variables.   The only symmetric *polynomials* in infinitely many variables are the constants.  To fix this, one allows infinitely many terms, as long as the *degree* is finite.

For example, there is a $1$-dimensional space of homogeneous symmetric functions of degree $1$, with basis

$$ x_1 + x_2 + \cdots = \sum_i x_i .$$

There is a $2$-dimensional space of homogeneous symmetric functions of degree $2$, with basis

$$ x_1^2 + x_2^2 + \cdots = \sum_i x_i^2 $$

and

$$ \sum_{i \lt j} x_i x_j .$$

These basis elementa are called the _[[elementary symmetric functions]]_.

(The homogeneous symmetric functions of degree $0$ are just the constants, as usual.)

There is also a noncommutative analogue: [[noncommutative symmetric functions]].

## Definitions

Let $\Lambda_n$ be the [[ring]] consisting of [[polynomials]] in $n$ [[variables]] $x_1, \dots, x_n$ that are invariant under all [[permutations]] of the variables; these are the **symmetric functions in $n$ variables** or **symmetric polynomials** in $n$ variables.   The rings $\Lambda_n$ are [[graded ring|graded]] by degree in the usual way, and there are homomorphisms of graded rings

$$ \Lambda_{n+1} \hookrightarrow \Lambda_{n} ,$$

given by setting the $(n+1)$st variable equal to zero.  Taking the [[limit]] (in the category-theoretic sense) of these rings $\Lambda_n$ as $n \to \infty$, we get the graded ring $\Lambda_\infty$.   This is usually called $\Lambda$, or the ring of **symmetric functions in countably many variables**, or simply **symmetric functions** (or even **symmetric polynomials**, even though very few of them are really polynomials).

Alternatively, $\Lambda$ can be constructed as a [[colimit]] of rings using the homomorphisms

$$ \Lambda_{n} \hookrightarrow \Lambda_{n+1} ,$$

where we add in new terms with the new variable to make the result symmetric.

The definition depends on the [[ground field|ground]] [[field]] (or [[commutative ring]] or [[rig]]) $k$, so we may write $\Lambda(k)$ to be precise.

A query about the Hazewinkel's description of the construction of $\Lambda$ as a (co)limit is [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=3117&Focus=25812#Comment_25812).

## Properties

Symmetric functions play a fundamental role throughout [[representation theory]], [[combinatorics]] and [[algebraic topology]].  The ring of symmetric functions, $\Lambda$, has many interesting properties.   For example, it is the free $\lambda$-[[lambda-ring|ring]] on one generator (a coincidence of notation that has not been ignored).  It is also a [[plethory]].  There are various bases of $\Lambda$ whose elements are in natural one-to-one correspondence with [[Young diagram|Young diagrams]].

$\Lambda$ acquires many of these properties from the fact that it is the [[Grothendieck ring]] of the category of $k$-linear [[species]].  This category is defined to be the functor category

$$ [\mathbb{P}, Vect_{k}] $$

where $\mathbb{P}$ is the groupoid of [[finite sets]] and bijections, and $Vect_k$ is the category of vector spaces over a field $k$ of characteristic zero.  The category of $k$-linear species becomes a symmetric monoidal category thanks to [[Day convolution]].   It is also a [[semisimple category|semisimple abelian category]], with a basis of objects given by irreducible [[representation|representations]] of [[symmetric groups]] --- one for each [[Young diagram]].  So, the [[Grothendieck group]] of this category becomes a commutative ring with a basis given by Young diagrams, and this is just $\Lambda$.

$\Lambda$ is also the Grothendieck ring of a somewhat smaller
$Schur$, whose objects are [[Schur functors]].    There are various ways to think about these, but they can be identified with $k$-linear species

$$ F: \mathbb{P} \to Vect_{k} $$

with the special property that $F(n)$ is finite-dimensional for all $n$ and $0$ for $n$ sufficiently large.   The category of Schur functors is again a [[semisimple category|semisimple abelian category]] with a basis of objects given by irreducible [[representation|representations]] of [[symmetric groups]], so its Grothendieck ring is again $\Lambda$.   For more on this, see [[Schur functor]].

\linebreak

The classification described above of irreducible $S_n$-modules over $\mathbb{C}$ also works unchanged for any algebraically closed field $k$ of characteristic zero.  It also works when $k$ has characteristic $p$ and $n$ is not divisible by $p$.   There\'s an exercise at the end of section 6.1 of Serre\'s book (page 64 of the french edition) that says that if $k$ is a field of characteristic $p \gt 0$ then the group algebra of the group $G$ is semisimple iff $p$ doesn\'t divide the order of $G$.

Apart from this, the field matters a lot.  There is a construction that gives all irreducible $k S_n$-modules for any field $k$, field, completely analogous to the Specht module construction over $\mathbb{C}$.  However, it describes each module as a quotient module of a Specht module, and in general not even the dimension of these irreducible modules is known, let alone an explicit basis, or representing matrices.

## Examples

* In the context of [[universal characteristic classes]] of [[vector bundles]], the [[splitting principle]] identifies higher [[Chern classes]] with symmetric polynomials in copies of the [[first Chern class]].

## Related concepts

* [[Witt vector]]

* [[Schur polynomial]]

* [[Schur positivity]]

## References

* [[Richard Stanley]], Section II of: *Theory and application of plane partitions 1*, Studies in Applied Math. **50** 2 (1971), 167-188 ([pdf](http://www-math.mit.edu/~rstan/pubs/pubfiles/12-1.pdf), [[StanleyPlanePartitions1.pdf:file]])

Among the best books in the subject are:

* [[Gordon D. James]], *The Representation Theory of the Symmetric Groups*,  Lecture Notes in Mathematics, volume 682, Springer 1978 ([doi:10.1007/BFb0067708](https://link.springer.com/book/10.1007/BFb0067708), [pdf](http://www-users.math.umn.edu/~webb/oldteaching/Year2010-11/the-representation-theory-of-the-symmetric-groups-SLN.pdf))

* Gordon James, Adalbert Kerber, _The representation theory of the symmetric group_, With a foreword by P. M. Cohn. With an intr. by G. de B. Robinson. Enc. of Math. and its Appl. __16__, Addison-Wesley 1981. xxviii+510 pp.

James also has a readable survey article that outlines developments in the '80's and '90's:

* Gordon D. James, Symmetric groups and Schur algebras, in _Algebraic Groups and their Representations_, eds. R. W. Carter and J. Saxl, Kluwer Acad. Publ. Netherlands 1998.

Other textbooks:

* {#Macdonald95} [[Ian G. Macdonald]], _Symmetric functions and Hall polynomials_, Oxford Math. Monographs, 2nd enlarged ed. 1995 ([ISBN:9780198739128](https://global.oup.com/academic/product/symmetric-functions-and-hall-polynomials-9780198739128?cc=ae&lang=en&))

* {#Stanley99} [[Richard Stanley]], Section 7 of: *Enumerative combinatorics 2*, Cambridge University Press (1999, 2010) ([doi:10.1017/CBO9780511609589](https://doi.org/10.1017/CBO9780511609589), [webpage](http://www-math.mit.edu/~rstan/ec/))

Another approach is described here:

* Alexander Kleshchev, _Linear and Projective Representations of Symmetric Groups_, Cambridge University Press, Cambridge 2009.

One can study symmetric functions in any characteristic, or over any integral domain.  The power sum symmetric functions do not generate the ring of symmetric functions over $\mathbb{Z}$, and this difference matters. They appear to be of limited usefulness in the description of the modular irreducible representations of $S_n$.

The following article surveys the subject in light of the connections to [[Hopf algebras]] and also to  [[noncommutative symmetric function|noncommutative analogue]]:

* [[Michiel Hazewinkel]], _Symmetric functions, noncommutative symmetric functions and quasisymmetric functions_, [pdf](http://oai.cwi.nl/oai/asset/10344/10344B.pdf)

See also

* [[Michiel Hazewinkel]], _[[Hazewinkel, Witt vectors|Witt vectors]]_, Part 1. [arXiv/0804.3888](http://arxiv.org/abs/0804.3888)

[[!redirects symmetric function]]
[[!redirects symmetric functions]]
[[!redirects symmetric polynomial]]
[[!redirects symmetric polynomials]]


symmetric group

+-- {: .rightHandSide}
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#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

The [[permutations]] of a [[set]] $X$ form a [[group]], $S_X$, under [[composition]].  This is especially clear if one thinks of the permutation as a [[bijection]] on $X$, where the multiplication / composition is just composition of [[functions]]. This group is called the _symmetric group_ of $X$ and often denoted $S_X$, $\Sigma_X$, $Sym(X)$ or similar.

The [[subgroups]] of symmetric groups are the _[[permutation groups]]_.

When $X$ is [[the]] [[finite set]] $(n) = \{1,\dots,n\}$, then its symmetric group is a [[finite group]] of [[cardinality]] $n!$ = "$n$ [[factorial]]", and one typically writes $S_n$ or $\Sigma_n$.

(We will make frequent use of the entry [[permutation]] for some key notation and concepts.)

In general, an element of $S_n$ can be represented by a two row array

$$\sigma= \left(\array{1&2&\ldots &n\\\sigma(1)&\sigma(2)&\ldots&\sigma(n)}\right).$$

It can also be given in cycle notation. We repeat the example from the entry on [[permutations]]. This is from $S_6$:

Let $\sigma$ be the permutation on $(6)$ defined by

$$\sigma = (\array{1 \mapsto 1 & 2\mapsto 4 & 3 \mapsto 5 & 4 \mapsto 6 & 5 \mapsto 3 & 6 \mapsto 2})= \left(\array{1&2&3&4&5&6\\1&4&5&6&3&2}\right)$$

The domain of the permutation is partitioned into three $\langle\sigma\rangle$-orbits

$$(6) = \{1\} \cup \{2,4,6\} \cup \{3,5\}$$

corresponding to the three cycles

$$1 \underset{\sigma}{\to} 1 \qquad
2 \underset{\sigma}{\to} 4 \underset{\sigma}{\to} 6 \underset{\sigma}{\to} 2 \qquad
3 \underset{\sigma}{\to} 5 \underset{\sigma}{\to} 3$$

We express this more compactly by writing $\sigma$ as the composition $\sigma = (1)(2\,4\,6)(3\,5)$, or $\sigma = (2\,4\,6)(3\,5)$ leaving implicit the action of the identity (1).

We can also write $\sigma$ as a product of transpositions

$$(2\,4)(2\,6)(3\,5)$$

## Examples

### The symmetric group  $S_3$

$S_3$ is the group of permutations of $(3)=\{1,2,3\}$.

One simple way to represent an element of $S_3$ is by listing the $(3)$ on the top row of an array with a second row denoting the image of each element under the permutation.  For instance, the element $sigma= (1\mapsto 2, 2\mapsto 1, 3\mapsto 3)$ can be written more compactly as

$$\sigma= \left(\array{1&2&3\\2&1&3}\right).$$

We can also use, even more compactly, 'cycle notation', as explained in more detail at [[permutation]], in which successive images under the permutation, $\sigma$, are listed until you get back to where you started. Elements of (3), or more generally of (n),  are usually not listed as cycles of length 1, exept that (1) may be used to indicate the identity element. For the above permutation, this gives

$$\sigma = (1\,2)$$

the _transposition_ exchanging the elements 1 and 2 and leaving 3 'unmoved'.  Whilst the 3-cycle, $(1\,2\,3)$, shifts every element to the right one position and then maps 3 to 1.

\begin{example}\label{CayleyGraphOfSym3}
**([[Cayley graph of Sym(3)]])**
The following shows the [[Cayley graph]] of the [[symmetric groups]] on 3 elements, $Sym(3)$, with [[edges]] corresponding to any [[transposition]] (not necessarily adjacent), hence whose [[graph distance]] is the [[Cayley distance]]:

\begin{tikzcd}[row sep=-6pt, column sep=14pt]
  123
  \ar[rrrr,-]
  \ar[ddddd,-]
  \ar[ddrrr,-]
  &&
  &&
  132
  \ar[ddddd,-]
  \\
  {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A}}}}}}}}
  \\
  &&&
  \scalebox{.8}{$321$}
  \ar[dddr,-]
  \\
  &
  \scalebox{1.2}{231}
  \ar[ddl,-]
  \ar[uuurrr,-, crossing over]
  \ar[urr,-]
  \\
  {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A} \atop {\phantom{A}}}}}}}}
  \\
  213
  \ar[rrrr,-]
  &&&&
  312
\end{tikzcd}

\end{example}

### The symmetric group $S_4$

The symmetric group on 4 elements is [[isomorphism|isomorphic]] to the full [[tetrahedral group]] as well as to the orientation-preserving [[octahedral group]].

## Properties

### Cycle decomposition

As an element of the [[symmetric group]] $S_X$, every permutation $\sigma : X \to X$ generates a [[cyclic group|cyclic]] [[subgroup]] $\langle \sigma \rangle$, and hence inherits a [[group action]] on $X$.  The [[orbits]] of this action partition the set $X$ into a disjoint union of cycles, called the **cyclic decomposition** of the permutation $\sigma$.

For example, let $\sigma$ be the permutation on $(6)$ defined by

$$\sigma = (\array{1 \mapsto 1 & 2 \mapsto 4 & 3 \mapsto 5 & 4 \mapsto 6 & 5 \mapsto 3 & 6 \mapsto 2})$$

The domain of the permutation is partitioned into three $\langle\sigma\rangle$-orbits

$$(6) = \{1\} \cup \{2,4,6\} \cup \{3,5\}$$

corresponding to the three cycles

$$1 \underset{\sigma}{\to} 1 \qquad
2 \underset{\sigma}{\to} 4 \underset{\sigma}{\to} 6 \underset{\sigma}{\to} 2 \qquad
3 \underset{\sigma}{\to} 5 \underset{\sigma}{\to} 3$$

We can express this more compactly by writing $\sigma$ in "cycle notation", as the composition $\sigma = (1)(2\,4\,6)(3\,5)$, or $\sigma = (2\,4\,6)(3\,5)$ leaving implicit the action of the identity (1).

Even in the simple example of $S_3$ one can see some patterns.

* $(12)$ clearly has order 2, whilst $(123)$ has square $(132)$, which is also the inverse of $(123)$, and has order 3.

* $(123) = (12)(13)$, so transpositions generate the whole group.

This gives a listing of the elements of $S_3$ as

$$\{ 1, (12),(13),(23),(123),(132)\}.$$

* $S_3$ has a [[group presentation|presentation]]

$$\langle a,b | a^3, b^2,(a b)^2 \rangle.$$

where $a$ corresponds to the 3-cycle, $(123)$, whilst $b$ corresponds to any transposition.  It also has a presentation in which the generators correspond to the transpositions:

$$\langle \sigma_1,\sigma_2 | \sigma_1^2,\sigma_2^2, (\sigma_1\sigma_2)^3\rangle$$

It is easy to see that these relations imply that

$$\sigma_1\sigma_2\sigma_1= \sigma_2\sigma_1\sigma_2$$

which relates this symmetric group to the [[braid group]], **Br 3**, and to the [[trefoil knot|trefoil group]].  It is clear that there is an epimorphism $S_3\to Br_3$ obtained by making the two braid generators become the transpositions.

### Conjugacy classes
 {#ConjugacyClasses}

+-- {: .num_prop #ConjugacycClassesOfSymmetricGroupCorrespondToCycleSet}
###### Proposition
**([[conjugacy classes]] of the [[symmetric group]] labeled by [[partitions]] by [[cycle]] lengths)** \linebreak

Let $n \in \mathbb{N}$ and $\Sigma(n)$ the [[symmetric group]]
on $n$ elements. Then the [[conjugacy classes]] of elements of $\Sigma(n)$, hence of [[permutations]] of $n$ elements, correspond to the [[cycle]] structures: two elements are conjugate to each other precisely if they have the same number $\# cycles \in \mathbb{N}$ of distinct [[cycles]] of the same lengths $(l_1 \geq l_2 \geq \cdots \geq l_{\#cycles})$, hence equivalently if they define the same underlying [[partition]]

$$
  l_1 + l_2 + \cdots + l_{\# cycles}
  \;=\;
  n
$$

of $n$.

=--

(e.g. [Sagan 01, p. 3 (18 of 254)](#Sagan01))

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

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

(1 2 3) ~ (1 3 2)

(1 2)(3) ~ (1 3)(2) ~ (1)(2 3)

(1)(2)(3)

 <svg width="80" height="40" xmlns="http://www.w3.org/2000/svg" se:nonce="39384" xmlns:se="http://svg-edit.googlecode.com" xmlns:xlink="http://www.w3.org/1999/xlink">
 <desc>Young diagram (3)</desc>
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  <title>Layer 1</title>
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 </g>
 </svg>

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  <rect x="10" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_3"/>
 </g>
 </svg>

 <svg width="40" height="90" xmlns="http://www.w3.org/2000/svg" se:nonce="39384" xmlns:se="http://svg-edit.googlecode.com" xmlns:xlink="http://www.w3.org/1999/xlink">
 <desc>Young diagram (1,1,1)</desc>
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  <title>Layer 1</title>
  <rect x="10" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_1"/>
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  <rect x="10" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_3"/>
 </g>
 </svg>

=--

$\,$

$\,$

$\,$

$\,$

$\,$

$\,$

---

### Representation theory

See at *[[representation theory of the symmetric group]]*.

### Relation to the field with one element

One may regard the symmetric group $S_n$ as the [[general linear group]] in dimension $n$ on the [[field with one element]]  $GL(n,\mathbb{F}_1)$.

See [there](field+with+one+element#AlgebraOverF1).

### Classifying space and universal Thom space
 {#ClassifyingSpaceAndThomSpace}

The [[classifying space]] $B \Sigma(n)$ of the symmetric group on $n$ elements may be presented by $Emb(\{1,\cdots, n\}, \mathbb{R}^\infty)/\Sigma(n)$, the [[Fadell's configuration space]] on $n$ unordered points in $\mathbb{R}^\infty$.

(e.g. [Bödigheimer 87, Example 10](#Boedigheimer87))

Write $\tau_n$ for the [[rank]] $n$ [[vector bundle]] over this which exhibits the canonical [[action]] of $\Sigma(n)$ on $\mathbb{R}^n$, by permutation of [[coordinates]].

The [[Thom space]] $B \Sigma(n)^{\tau_n}$ of this bundle appears as the cofficients of the [[spectral symmetric algebra]] of the "absolute [[spectral superpoint]]" $Sym_{\mathbb{S}} \Sigma \mathbb{S}$ (see [Rezk 10, slide 4](spectral%20symmetric%20algebra#Rezk10)).

See also ([Hopkins-Mahowald-Sadofsky 94, around def. 2.8](#HopkinsMahowaldSadofsky94))

### Whitehead tower and relation to supersymmetry
 {#WhiteheadTowerAndSupersymmetry}

The [[symmetric groups]] and [[alternating groups]] are the first stages in a restriction of the [[Whitehead tower]] of the [[orthogonal group]] to "finite [[discrete ∞-groups]]" in the sense of [[homotopy type with finite homotopy groups]]. The [[homotopy fibers]] of the stages of the "finite Whitehead tower" are the [[stable homotopy groups of spheres]] ([Epa-Ganter 16](#EpaGanter16)). (See also at _[super algebra -- Abstract idea](super+algebra#AbstractIdea)_ and at _[[Platonic 2-group]]_.)

$$
  \array{
    && && \vdots
    \\
    && && \downarrow
    \\
    && && \mathbf{B}Fivebrane(n)
    \\
    && \downarrow && \downarrow
    \\
    \mathbf{B}^2 \pi_3 \mathbb{S}
      &\stackrel{}{\longrightarrow} &
     \mathbf{B}\mathcal{A}_n &\hookrightarrow& \mathbf{B} String(n)
    \\
    && \downarrow && \downarrow
    \\
    \mathbf{B} \pi_2 \mathbb{S} &\longrightarrow & \mathbf{B} \tilde A_n &\hookrightarrow& \mathbf{B} Spin(n)
    \\
    && \downarrow && \downarrow
    \\
    \pi_1 \mathbb{S} &\longrightarrow& \mathbf{B} A_n &\hookrightarrow& \mathbf{B} SO(n)
    \\
    && \downarrow && \downarrow
    \\
    && \mathbf{B} S_n &\hookrightarrow& \mathbf{B} O(n)
  }
$$

* on the right: the [[delooping|delooped]] [[smooth ∞-group]] [[Whitehead tower]] of the [[orthogonal group]] ([[fivebrane 6-group]] $\to$ [[string 2-group]] $\to$ [[spin group]] $\to$ [[special orthogonal group]] $\to$ [[orthogonal group]]);

* in the middle, its restriction to [[deloopings]] of [[finite groups]] and their universal [[∞-group extensions]] ($\cdots \to $ [[covering of alternating group]] $\to$ [[alternating group]] $\to$ [[symmetric group]])

* on the left the [[homotopy fibers]] of each stage.

Notice that the squares on the right are _not_ [[homotopy pullback]] squares. (The homotopy pullback of the [[string 2-group]] along $\tilde A \hookrightarrow Spin(n)$ is a $\mathbf{B}U(1)$-extension of $\tilde  A$, but here we get the universal _finite_ 2-group extension, by $\mathbb{Z}/24$ instead.

### Cardinality of symmetric groups

Recall that two sets $A$ and $B$ have the same [[cardinality]] if $A$ is in [[bijection]] with $B$.

Given a [[natural number]] $n:\mathbb{N}$, the symmetric group on [[generalized the|the]] [[finite set]] with $n$ elements $\Sigma(n)$ is in bijection with the [[finite set]] $\mathrm{Fin}(n!)$ elements, where $n!$ is the [[factorial]] of $n$.

The symmetric group on the natural numbers $\mathrm{Sym}(\mathbb{N})$ is an [[uncountable set]] in [[bijection]] with the endofunction set $\mathbb{N}^\mathbb{N}$. In the context of [[excluded middle]], this is in bijection with $\mathcal{P}(\mathbb{N})$, the [[power set]] of the natural numbers.

Assuming the [[axiom of choice]], given an [[infinite set]] $A$, there is a [[bijection]] $\mathrm{Sym}(A) \cong \mathcal{P}(A)$ between the symmetric group of $A$ and the [[power set]] of $A$.

## Related concepts

* [[permutation]], [[permutation representation]], [[permutation matrix]]

* [[bijection set]]

* [[Cayley distance]], [[Kendall distance]]

  [[Cayley distance kernel]], [[Mallows kernel]]

* [[braid group]]

* [[alternating group]]

* [[symmetric groupoid]]

* [[signature of a permutation]]

* [[Young subgroup]]

* [[permutation D-brane]]

* [[automorphism group]], [[automorphism infinity-group]]

* [[autoequivalence type]]

## References

### General

Textbook accounts:

* {#Sagan01} [[Bruce Sagan]], _The symmetric group_, Springer 2001 ([doi:10.1007/978-1-4757-6804-6](https://link.springer.com/book/10.1007/978-1-4757-6804-6), [pdf](http://math.sfsu.edu/federico/Clase/RepTh/sagan.pdf))

Discussion of the [[classifying spaces]] of symmetric groups:

* {#Boedigheimer87} [[Carl-Friedrich Bödigheimer]], _Stable splittings of mapping spaces_, Algebraic topology. Springer 1987. 174-187 ([pdf](http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf))

See also

* Wikipedia, _[Symmetric group](https://en.wikipedia.org/wiki/Symmetric_group)_

* {#EpaGanter16} [[Narthana Epa]], [[Nora Ganter]], _Platonic and alternating 2-groups_, Higher Structures 1(1):122-146, 2017 ([arXiv:1605.09192](http://arxiv.org/abs/1605.09192))

* {#HopkinsMahowaldSadofsky94} [[Michael Hopkins]], [[Mark Mahowald]], [[Hal Sadofsky]], _Constructions of elements in Picard groups_, Contemporary mathematics, Volume 158, 1994 in [[Eric Friedlander]], [[Mark Mahowald]] (eds.), _Topology and Representation theory_ ([doi:10.1090/conm/158/01454](http://dx.doi.org/10.1090/conm/158/01454))

For the operadic structure of permutations:

* {#FresseHOGTG} [[Benoit Fresse]], Volume 1, Chapter 1 of: _Homotopy of Operads and Grothendieck-Teichmüller Groups (Volumes 1 & 2)_, [website](http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/)

For more on permutation patterns, see:

* Wikipedia, [Permutation pattern](https://en.wikipedia.org/wiki/Permutation_pattern).

* Étienne Ghys, _A Singular Mathematical Promenade_, August 2017. [arXiv:1612.06373](https://arxiv.org/abs/1612.06373) [author pdf](http://perso.ens-lyon.fr/ghys/promenade/)

That in the context of the [[axiom of choice]], the [[symmetric group]] of an [[infinite set]] is in [[bijection]] (i.e. has the same [[cardinality]]) with the [[power set]] of that infinite set, see:

* John Dawson; Paul Howard, *Factorials of infinite cardinals*, Fundamenta Mathematicae (1976), Volume: 93, Issue: 3, page 185-195, ISSN: 0016-2736 ([EuDML](https://eudml.org/doc/214975))

[[!redirects symmetric groups]]

[[!redirects Sym(n)]]


symmetric matrix

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#### Linear algebra
+-- {: .hide}
[[!include higher linear algebra - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

A square [[matrix]] $A$ is called _symmetric_ if it is [[equality|equal]] to its own [[transpose matrix]]: $A = A^T$.

Symmetric matrices correspond to [[symmetric bilinear forms]]. Accordingly a symmetric matrix is called a positive or negative (semi-)definite matrix if the corresponding bilinear form is such (see [there](inner+product+space#definite)).

## Related concepts

* [[skew-symmetric matrix]]

* [[hermitean matrix]]

* [[triangular matrix]]

## References

See also

* Wikipedia, _[Symmetric matrix](https://en.wikipedia.org/wiki/Symmetric_matrix)_

* Wikipedia, _[Positive-definite matrix](https://en.wikipedia.org/wiki/Positive-definite_matrix)_

[[!redirects symmetric matrices]]

[[!redirects positive-definite matrix]]
[[!redirects positive-definite matrices]]

[[!redirects positive definite matrix]]
[[!redirects positive definite matrices]]

[[!redirects negative-definite matrix]]
[[!redirects negative-definite matrices]]

[[!redirects negative definite matrix]]
[[!redirects negative definite matrices]]

[[!redirects positive-semidefinite matrix]]
[[!redirects positive-semidefinite matrices]]

[[!redirects positive semidefinite matrix]]
[[!redirects positive semidefinite matrices]]

[[!redirects negative-semidefinite matrix]]
[[!redirects negative-semidefinite matrices]]

[[!redirects negative semidefinite matrix]]
[[!redirects negative semidefinite matrices]]

[[!redirects indefinite matrix]]
[[!redirects indefinite matrices]]

[[!redirects positive-definite symmetric matrix]]
[[!redirects positive-definite symmetric matrices]]

[[!redirects positive definite symmetric matrix]]
[[!redirects positive definite symmetric matrices]]

[[!redirects negative-definite symmetric matrix]]
[[!redirects negative-definite symmetric matrices]]

[[!redirects negative definite symmetric matrix]]
[[!redirects negative definite symmetric matrices]]

[[!redirects positive-semidefinite symmetric matrix]]
[[!redirects positive-semidefinite symmetricmatrices]]

[[!redirects positive semidefinite symmetric matrix]]
[[!redirects positive semidefinite symmetric matrices]]

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[[!redirects negative-semidefinite symmetric matrices]]

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[[!redirects indefinite symmetric matrices]]


symmetric midpoint algebra

+-- {: .rightHandSide}
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#### Algebra
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#Contents#
* table of contents
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## Idea

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

## Definition

A __symmetric midpoint algebra__ is a [[midpoint algebra]] $(M,\vert)$ with an element $\odot:M$ and a function $(-)^{\bullet}: M \to M$ such that

* for all $a$ in $M$, $(a^{\bullet})^{\bullet} = a$

* for all $a$ and $b$ in $M$, $a^{\bullet} \vert a = \odot$

* for all $a$ and $b$ in $M$, $(a \vert b)^{\bullet} = a^{\bullet} \vert b^{\bullet}$

## Properties

$\odot$ is the only element in $M$ such that $\odot^\bullet = \odot$.

## Examples

The [[rational numbers]], [[real numbers]], and the [[complex numbers]] with $a \vert b \coloneqq \frac{a + b}{2}$, $\odot = 0$, and $a^{\bullet} = -a$ are examples of symmetric midpoint algebras.

The [[trivial group]] with $a \vert b = a \cdot b$, $\odot = 1$ and $a^{\bullet} = a^{-1}$ is a symmetric midpoint algebra.

## Related concepts

* [[midpoint algebra]]

* [[cancellative midpoint algebra]]

* [[symmetric cancellative midpoint algebra]]

* [[symmetric midpoint group]]

* [[dyadic rational 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))


symmetric midpoint group

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#Contents#
* table of contents
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## Idea

A [[symmetric midpoint algebra]] that is also a [[0-truncated]] [[symmetric 2-group]].

## Definition

A __symmetric midpoint group__ or __abelian midpoint group__ is a set $G$ with an element $0:G$, a function $-: G \to G$, a binary operation $(-)+(-):G \times G \to G$ and a binary operation $(-)\vert(-): G \times G \to G$ such that

* $(G,+,0,-)$ is an [[abelian group]]

* $(G,\vert,0,-)$ is a [[symmetric midpoint algebra]]

* for all $a$ and $b$ in $G$, $a \vert b + a \vert b = a + b$

## Examples

The [[rational numbers]], [[real numbers]], and the [[complex numbers]] with $a \vert b \coloneqq \frac{a + b}{2}$ are examples of symmetric midpoint groups.

The [[dyadic rational numbers]] are the [[free object|free]] symmetric midpoint group on one [[generator]].

The [[trivial group]] is a symmetric midpoint group, and is in fact a [[zero object]] in the [[category]] of symmetric midpoint groups.

## Related concepts

* [[abelian group]]

* [[symmetric midpoint algebra]]

* [[dyadic rational number]]


+-- {: .rightHandSide}
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# Contents
* table of contents
{: toc}

##Idea#

A _symmetric monoidal $(\infty,1)$-category_ is

* an [[(∞,1)-category]]

* which is "$\infty$-[[k-tuply monoidal n-category|tuply monoidal]]", or "stably monoidal".

This means that it is

* a [[monoidal (∞,1)-category]];

* for which the [[tensor product]] is commutative up to infinite coherent homotopy.

This can be understood as a special case of an [[(∞,1)-operad]] (...to be expanded on...)

Equivalently, a symmetric monoidal $(\infty,1)$-category is a [[commutative algebra in an (infinity,1)-category]] in the [[(infinity,1)-category of (infinity,1)-categories]].

Just as many ordinary $(\infty,1)$-categories (particularly, all of those that are [[locally presentable (infinity,1)-category|locally presentable]]) can be presented by [[model categories]], many symmetric monoidal $(\infty,1)$-categories can be presented by symmetric [[monoidal model categories]].  See for instance [NikolausSagave15](#NikolausSagave15).

## Definition in terms of quasi-categories

Recall that in terms of [[quasi-category|quasi-categories]] a general [[monoidal (infinity,1)-category]] is conceived as a coCartesian fibration $C^\otimes \to N(\Delta)^{op}$ of [[simplicial set]]s over the ([[opposite category|opposite]] of) the [[nerve]]  $N(\Delta)^{op}$ of the [[simplex category]] satisfying a certain property.

The fiber of this fibration over the 1-[[simplex]] $[1]$ is the [[monoidal (infinity,1)-category]] $C$ itself, its value over a map $[n] \to [1]$ encodes the tensor product of $n$ factors of $C$ with itself.

The following definition encodes the _commutativity_ of all these operations by replacing $\Delta$ with the category $FinSet_*$ of pointed finite sets.

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

A **symmetric monoidal $(\infty,1)$-category** is a
[[coCartesian fibration]] of [[simplicial set]]s

$$
  p : C^\otimes \to N(FinSet_*)
$$

such that

* for each $n \geq 0$  the associated functors $C^\otimes_{[n]} \to C^\otimes_{[1]}$ determine an equivalence of $(\infty,1)$-categories $C^\otimes_{[n]} \stackrel{\simeq}{\to} C_{[1]}^n$.

=--

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

In other words, a symmetric monoidal $(\infty,1)$-category is an
$\mathcal{O}$-[[monoidal (∞,1)-category]] for

$$
  \mathcal{O} = Com
$$

the [[commutative operad|commutative]] [[(∞,1)-operad]].

=--

See ([Lurie, def. 2.0.0.7](#LurieAlgebra)).

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

  The [[homotopy category of an (infinity,1)-category|homotopy category]] of a symmetric monoidal $(\infty,1)$-category is an ordinary [[symmetric monoidal category]].

=--

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

There is a functor $\varphi : \Delta^{op} \to FinSet_*$ such that the [[monoidal (infinity,1)-category]] _underlying_ a symmetric monoidal $(\infty,1)$-category $p : C^\otimes \to N(FinSet_*)$ is the [[(infinity,1)-pullback]] of $p$ along $\varphi$.
=--

## Examples
 {#Examples}

### Classes of examples

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

### Specific examples

* [[stable (infinity,1)-category of spectra]]

* [[symmetric monoidal (infinity,1)-category of presentable (infinity,1)-categories]]

## Properties

### Model category structure

A [[presentable (infinity,1)-category|presentation]] of the [[(∞,1)-category]] of all symmetric monoidal $(\infty,1)$-categories is provided by the [[model structure for dendroidal coCartesian fibrations]].

### Commutative $\infty$-monoids

See [[commutative monoid in a symmetric monoidal (∞,1)-category]].

## Related concepts

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

* [[symmetric monoidal category]], **symmetric monoidal $(\infty,1)$-category**, [[symmetric monoidal (∞,n)-category]]

* [[tensor (∞,1)-category]]

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

* [[K-theory of a symmetric monoidal (∞,1)-category]]

* [[prime spectrum of a symmetric monoidal stable (∞,1)-category]]

## References

The defintion of symmetric monoidal quasi-category is definition 1.2 in

* [[Jacob Lurie]], _[[Commutative Algebra]]_

and definition 2.0.0.7 in

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#LurieAlgebra}

A concise treatment is also available in

* [[Moritz Groth]], _A short course on infinity-categories_, [pdf](http://www.math.ru.nl/~mgroth/preprints/groth_scinfinity.pdf).

Relation to [[monoidal model categories]] (in particular, that every locally presentable symmetric monoidal $(\infty,1)$-category arises from a symmetric monoidal model category) is discussed in

* {#NikolausSagave15} [[Thomas Nikolaus]], [[Steffen Sagave]], _Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories_. Algebr. Geom. Topol. 17 (2017), no. 5, 3189--3212. ([arXiv:1506.01475](http://arxiv.org/abs/1506.01475))

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


symmetric monoidal (infinity,n)-category

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* table of contents
{:toc}

## Idea

A _symmetric monoidal $(\infty,n)$-category_ is the analog of a [[symmetric monoidal (∞,1)-category]] for [[(∞,n)-category]] theory.

##Definition

As [[Jacob Lurie]] says in [Lurie, Mathoverflow quote](#Lurie):

> {#Definition} There are many (equivalent) definitions for the notion of symmetric monoidal $(\infty,n)$-category. One approach is based on the observation that a monoidal category can be identified with a bicategory having only a single object. You can define a monoidal
$(\infty,n)$-category to be an $(\infty,n+1)$-category with a specified object,
such that all other objects are isomorphic to it (in the complete Segal space model, this means that the space of objects should be connected). Similarly you can define a braided monoidal $(\infty,n)$-category to be an $(\infty,n+2)$-category equipped with a distinguished object satisfying a simple connectivity condition, and so on and so forth. You get to the symmetric monoidal case by taking the homotopy inverse limit (that is, a symmetric monoidal
$(\infty,n)$-category is a collection of pointed $(\infty,n+k)$-categories, each of which is obtained by "looping" the next one). You might find this definition convenient in the context of bordism categories, since they are naturally related in this way (if you ``loop'' the a bordism category of $d$-manifolds, you get a bordism category of $d+1$-manifolds: and this is sensible even when $d$ is negative).

>Alternatively, you can define a symmetric monoidal $(\infty,n)$-category to be a commutative monoid in the setting of $(\infty,n)$-categories. There are many ways to formalize this. Since you're asking about specific models, let's suppose you start with some model category $\mathbf{A}$ for the homotopy theory of $(\infty,n)$-categories (higher-dimensional Segal spaces, for example). If you have a simplicial model category, you can do as Charles suggested and take algebras for some $E_{\infty}$-operad in simplicial sets. If you'd prefer not to mention operads, you can just copy Segal's definition of a $\Gamma$-space: take the category of functors $F$ from pointed finite sets into $\mathbf{A}$, and equip it with a model structure that enforces the relevant Segal condition.

>As Martin mentions in [his answer](http://mathoverflow.net/questions/81425/what-is-a-symmetric-monoidal-infty-n-category/81432#81432), there is an extensive discussion of the case $n=1$ in my book, and also of commutative monoids in an arbitrary $(\infty,1)$-category (of which this is a special case, since the collection of all $(\infty,n)$-categories can be regarded as an $(\infty,1)$-category). I don't know of references that address your question more specifically (though I would not be surprised if there were some).

## Properties

### Dualizable objects

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

An object $X$ in a symmetric monoidal $(\infty,n)$-category is called **dualizable** if ...

=--

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

Let $C$ be a symmetric monoidal $(\infty,n)$-category. Then there exists another symmetric monoidal $(\infty,n)$-category $C^{fd}$ and a symmetric monoidal functor

$$
  C^{fd} \to C
$$

such that $C^{fd}$ has duals and is [[universal property|universal]] with these properties:

for any symmetric monoidal [[(∞,n)-category with duals]] $D$ and any symmetric monoidal functor $F : D \to C$ there exists a symmetric monoidal functor $f : D \to C^{fd}$, unique up to equivalence, and an equivalence

$$
  \array{
     && C^{fd}
     \\
     & {}^{\mathllap{f}}\nearrow &\Downarrow^{\simeq}& \searrow
     \\
     D &&\stackrel{F}{\to}&& C
  }
  \,.
$$

=--

This appears as ([Lurie, claim 2.3.19](#Lurie)).

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

$C^{fd}$ is obtained from $C$ by discarding all objects that do not have duals and all [[k-morphism]]s that do not admit right and left adjoints.

=--

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

An object $X \in C$ is called a **[[fully dualizable object]]** if it is in the essential [[image]] of $C^{fd} \to C$.

=--

## Examples

* For all $n \in \mathbb{N}$, the [[(∞,n)-category of cobordisms]] $Bord_n$ is symmetric monoidal. By the [[cobordism hypothesis]] this should be the [[free functor|free]] [[symmetric monoidal (∞,n)-category]] [[(∞,n)-category with duals|with duals]] on the [[point]].

* For all $n$ and $C$ any symmetric monoidal $(\infty,n)$-category, there is a symmetric monoidal [[(∞,n)-category of spans]] of [[∞-groupoid]]s over $C$.

## Related concepts

* [[2d TQFT]], [[TCFT]]

  [[Calabi-Yau object]]

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

  * [[symmetric monoidal category]], [[symmetric monoidal (∞,1)-category]], **symmetric monoidal $(\infty,n)$-category**

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

* [[symmetric monoidal 2-category]]

* [[(∞,n)-category with adjoints]]

* [[(∞,n)-category with duals]]

## References

A discussion of dualizable objects is in section 2.3 of

* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_
{#Lurie}

* {#Lurie} [[Jacob Lurie]], [Mathoverflow/81425 answer](http://mathoverflow.net/questions/81425/what-is-a-symmetric-monoidal-infty-n-category/81445#81445).

[[!redirects symmetric monoidal (infinity,n)-categories]]

[[!redirects symmetric monoidal (∞,n)-category]]
[[!redirects symmetric monoidal (∞,n)-categories]]

[[!redirects monoidal (∞,n)-category]]
[[!redirects monoidal (infinity,n)-category]]
[[!redirects monoidal (∞,n)-categories]]
[[!redirects monoidal (infinity,n)-categories]]


symmetric monoidal (∞,2)-category

## Idea

[[symmetric monoidal (infinity,n)-category]] for $n = 2$.

## Related concepts

* [[Calabi-Yau object]]

* [[2d TQFT]], [[TCFT]]

[[!redirects symmetric monoidal (infinity,2)-category]]
[[!redirects symmetric monoidal (infinity,2)-categories]]


symmetric monoidal (∞,n)-category > history

> see [[symmetric monoidal (∞,n)-category]].


symmetric monoidal 2-category

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#Contents#
* table of contents
{:toc}

## Definition

A _symmetric monoidal (weak) 2-category_ is a [[monoidal 2-category]] with a categorified version of a [[symmetric monoidal category|symmetry]].

That is, it is a [[2-category]] $C$ equipped with a [[tensor product]] $\otimes : C \times C \to C$ [[2-functor]] which satisfies all possible conditions for being commutative up to equivalence.  In the language of [[k-tuply monoidal n-category|k-tuply monoidal n-categories]], a braided monoidal 2-category is a *quadruply monoidal 2-category*.  As described there, this may be identified with a pointed 6-category with a single $k$-morphism for $k=0,1,2,3$.  We can also say that it is a monoidal 2-category whose [[En-algebra|E1-algebra]] structure is refined to an [[En-algebra|E4-algebra]] structure.

## Examples

* A symmetric monoidal 2-category all whose objects are invertible under the tensor product is a [[symmetric 3-group]].

## Related concepts

* [[monoidal 2-category]], [[3-group]]

* [[braided monoidal 2-category]], [[braided 3-group]]

* [[sylleptic monoidal 2-category]], [[sylleptic 3-group]]

* **symmetric monoidal 2-category**, [[symmetric 3-group]]

* [[k-tuply monoidal (n,r)-category]]

## References

* {#DayStreet97} [[Brian Day]], [[Ross Street]], Section 5 of: *Monoidal Bicategories and Hopf Algebroids*, Advances in Mathematics Volume 129, Issue 1, 15 July 1997, Pages 99-157 ([doi:10.1006/aima.1997.1649](https://doi.org/10.1006/aima.1997.1649))

* [[Mike Shulman]], _Constructing symmetric monoidal bicategories_ ([arXiv:1004.0993v1](http://arxiv.org/abs/1004.0993v1))

[[!redirects symmetric monoidal 2-categories]]

[[!redirects symmetric monoidal bicategory]]
[[!redirects symmetric monoidal bicategories]]


symmetric monoidal category

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{:toc}

## Idea

A _symmetric monoidal category_ is a category with a product operation -- a [[monoidal category]] -- for which the product is _as commutative as possible_.

The point is that there are different degrees to which higher categorical products may be commutative. While a bare [[monoid]] is either commutative or not, a [[monoidal category]] may be a [[braided monoidal category]] -- which already means that the order of products may be reversed up to some [[isomorphism]] -- without being _symmetric_ monoidal -- which means that changing the order of a product twice, from $a \otimes b$ to $b \otimes a$ back to $a \otimes b$, indeed does yield a result [[equality|equal]] to the original.

For higher monoidal categories there are accordingly ever more shades of the notion of "commutativity" of the monoidal product. This is described in detail at [[k-tuply monoidal n-category]].

In general, the term _symmetric monoidal_ is used for the maximally commutative case. See for instance [[symmetric monoidal (∞,1)-category]]. Notably, a symmetric monoidal [[∞-groupoid]] is, under the [[homotopy hypothesis]], the same as a [[spectrum|connective spectrum]].

A symmetric monoidal category is a special case of the notion of [[symmetric pseudomonoid]] in a [[sylleptic monoidal 2-category]].

## Definition

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

A **symmetric monoidal category** is a [[braided monoidal category]] for which the [[braiding]]

$$
   B_{x,y} \colon x \otimes y \to y \otimes x
$$

satisfies the condition:

$$
  B_{y,x} \circ B_{x,y} = 1_{x \otimes y}
$$

for all objects $x, y$

=--

Intuitively this says that switching things twice _in the same direction_ has no effect.

Expanding this out a bit: a  **symmetric monoidal category** is, to begin with a [[category]] $M$ equipped with a [[functor]]
$$ \otimes : M \times M \to M $$
called the **tensor product**, an object
$$ 1 \in M $$
called the **unit object**, a natural isomorphism
$$ a_{x,y,z} : (x \otimes y) \otimes z \to x \otimes (y \otimes z) $$
called the **associator**, a natural isomorphism
$$ \lambda_x : 1 \otimes x \to x $$
called the **left unitor**, a natural isomorphism
$$  \rho_x : x \otimes 1 \to x $$
called the **right unitor**, and a natural isomorphism
$$ B_{x,y} : x \otimes y \to y \otimes x $$
called the **braiding**.  We then demand that the associator obey the **pentagon identity**, which says this diagram commutes:

+--{: style="text-align:center"}
[[!include monoidal category > pentagon]]
=--

We demand that the associator and unitors  obey the **triangle identity**, which says this diagram commutes:

$$
  \array{
     & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y)
      \\

      & {}_{\rho_x \otimes 1_y}\searrow

       && \swarrow_{1_x \otimes \lambda_y}
      &
     \\
     &&
     x \otimes y
     &&

  }
$$

We demand that the braiding and associator obey the first **hexagon identity**:

$$
  \array{
   (x \otimes y) \otimes z
   &\stackrel{a_{x,y,z}}{\to}&
   x \otimes (y \otimes z)
   &\stackrel{B_{x,y \otimes z}}{\to}&
   (y \otimes z) \otimes x
   \\
   \downarrow^{B_{x,y}\otimes 1_z}
   &&&&
   \downarrow^{a_{y,z,x}}
   \\
   (y \otimes x) \otimes z
   &\stackrel{a_{y,x,z}}{\to}&
   y \otimes (x \otimes z)
   &\stackrel{1_y \otimes B_{x,z}}{\to}&
   y \otimes (z \otimes x)
  }
$$

And lastly, we demand that
$$ B_{y,x} B_{x,y} = 1_{x \otimes y} . $$
(The definition of [[braided monoidal category]] has _two_ hexagon identities, but either one implies the other given this equation.)

## Properties

### The 2-category of symmetric monoidal categories

There is a [[strict 2-category]] $SymmMonCat$ with:

* symmetric monoidal categories as objects,
* [[symmetric monoidal functor]]s as morphisms,
* [[symmetric monoidal natural transformation]]s as 2-morphisms.

This 2-category has (weak) 2-[[biproducts]] given by the cartesian product of underlying categories (analogously to how [[Ab]] has biproducts given by the cartesian product of underlying sets).  For a proof, see [Fong-Spivak, Theorem 2.3](#FongSpivak), or for a more abstract version involving [[pseudomonoids]] [Schaeppi, Appendix A](#Schaeppi).

### As models for connective spectra
  {#ModelsForConnectiveSpectra}

The [[group completion]] of the [[nerve]] of a symmetric monoidal category is always an  [[infinite loop space]], hence the degree-0-space of a [[connective spectrum]]. One calls this also the [[K-theory]] spectrum of the symmetric monoidal category:

$$
  \array{
     &&&& Spectra
     \\
      && {}^{\mathllap{K}}\nearrow && \downarrow
    \\
    SymmMonCat &\to& Cat &\underset{N}{\to}& sSet
  }
  \,.
$$

This construction extended to an [[equivalence of categories]]

$$
  K :  Ho(SymmMonCat)
   \stackrel{\simeq}{\to}
  Ho(Spectra)_{\geq 0}
  \hookrightarrow
  Ho(Spectra)
$$

between the [[full subcategory]] of the [[stable homotopy category]] $Ho(Spectra)$ on the [[connective spectra]] and the [[homotopy category]] of $SymmMonCat$, regarded with the [[transferred model structure|transferred structure]] of a [[category with weak equivalences]].

This is due to ([Thomason, 95](#Thomason)). Further discussion is in ([Mandell, 2010](#Mandell)).

Notice that this is _almost_ the complete analog in [[stable homotopy theory]] of the [[Quillen equivalence]] between the [[Thomason model structure]] on [[Cat]] and the standard [[model structure on simplicial sets]]. Only that $SymmMonCat$ cannot carry a [[model category]] structure because it does not have all [[colimit]]s. In some sense the "colimit completion" of $SymmMonCat$ is the category of [[multicategories]]. Once expects that this carries a model structure that refines the above equivalence of homotopy categories to a [[Quillen equivalence]].

(This is currently being investigated by Elmendorf, Nikolaus and maybe others.)

However, a subcategory of $SymmMonCat$ whose objects are [[permutative categories]] and maps are symmetric strict monoidal functors, denoted by $Perm$  has a model category structure  which is transferred from the natural model category structure on $Cat$, see [Sharma](#Sharma). This model category structure is combinatorial, left-proper and a $Cat$-model category structure. It is referred to as the natural model category structure on $Perm$. The [[coherence theorem]] for symmetric monoidal categories states that each symmetric monoidal category is equivalent to a [[permutative category]].

### Relation to $\Gamma$-categories

The aforementioned natural model category of permutative categories is NOT a symmetric monoidal closed model category. This shortcoming was overcome in [Sharma] by constructing a Quillen equivalent [[model category]] which is symmetric monoidal closed.
A (unnormalized) $\Gamma$-category is a functor from $\Gamma^{op}$ to $Cat$, where $\Gamma^{op}$ is a skeletal category of finite based sets and based maps.
The category of $\Gamma$-categories and natural transformations, denoted by $\Gamma$$Cat$, is a symmetric monoidal closed category under the [[Day convolution]] product. The aforementioned symmetric monoidal closed model category is constructed in [Sharma](#Sharma) as a left-Bousfield localization of the projective model category structure on $\Gamma$$Cat$. A $\Gamma$-category is fibrant in this model category if it satisfies the Segal's condition in which case it is referred to as a coherently commutative monoidal category. The main result of [Sharma](#Sharma) is that an unnormalized version of the classical Segal's nerve functor is the right [[Quillen functor]] of a [[Quillen equivalence]] between the natural model category of permutative categories and the symmetric monoidal closed model category of coherently commutative monoidal categories.

### As algebras over the little $k$-cubes operad

A symmetric monoidal category is equivalently a category that is equipped with the structure of an [[algebra over an operad|algebra over]] the [[little cubes operad|little k-cubes operad]] for $k \geq 3$

Details are in examples 1.2.3 and 1.2.4 of

* [[Jacob Lurie]], $\mathbb{E}[k]$-[[Ek-Algebras|Algebras]]

### Tannaka duality

[[!include structure on algebras and their module categories - table]]

### Grothendieck ring

The [[Grothendieck group]] of a [[monoidal category]] naturally has the structure of a [[monoid]], of an [[abelian category|abelian]] monoidal category that of a [[ring]], of an abelian [[braided monoidal category]] that of a [[commutative ring]] and, finally, of an abelian symmetric monoidal category that of a _[[Lambda-ring]]_. See there for more.

### Internal logic

The [[internal logic]] of ([[closed monoidal category|closed]]) symmetric monoidal categories is called _[[linear logic]]_. This notably contains [[quantum logic]].

### Permutations

For every $n \ge 2$ and $\sigma \in \mathfrak{S}_{n}$, there is a [[natural transformation]]
$$\sigma_{A_{1},...,A_{n}}: A_{1} \otimes ... \otimes A_{n} \rightarrow A_{\sigma^{-1}(1)} \otimes ... \otimes A_{\sigma^{-1}(n)}$$
that is defined by the decomposition of a [[permutation]] in a product of adjacent [[transpositions]] and generalizes the [[braiding]]
$$
B:A_{1} \otimes A_{2} \rightarrow A_{2} \otimes A_{1}
$$

We recall that an adjacent transposition $[1,n] \rightarrow [1,n]$ where $n \ge 2$ is a permutation $t:[1,n] \rightarrow [1,n]$ such that there exists a $1 \le i \le n-1$ such that:
$$
t(1)=1,..., t(i-1)=i-1, t(i) = i+1, t(i+1) = i, t(i+2)=i+2,.... ,t(n)=n
$$
This $i$ is then unique, and for every $1 \le i \le n-1$, we note $t_{i}$ the associated adjacent transposition.

For $n \ge 2$, every permutation $\sigma \in \mathfrak{S}_{n}$ admits at least a decomposition under the form $\sigma = t_{i_{1}}; ...;t_{i_{q}}$
where $q \ge 0$, and $1 \le i_{1},....,i_{q} \le n-1$.

In every symmetric monoidal category, for $n \ge 2$ and $1 \le i \le n-1$, we have a natural transformation
$$t_{i}:A_{1} \otimes .... \otimes A_{n} \rightarrow A_{1} \otimes ... \otimes A_{i-1} \otimes A_{i+1} \otimes A_{i} \otimes A_{i+2} \otimes .... \otimes A_{n}$$
defined by:
$$t_{i} = 1_{A_{1}} \otimes ... \otimes 1_{A_{i-1}} \otimes \B_{A_{i},A_{i+1}} \otimes 1_{A_{i+2}} \otimes ... \otimes 1_{A_{n}}$$

Given a $\sigma \in \mathfrak{S}_{n}$, the associated natural transformation is then defined as $t_{i_{1}};....;t_{i_{q}}$ for any such decomposition of $\sigma$. The fact that the result doesn't depend on the particular decomposition is a consequence of the [[coherence theorem for symmetric monoidal categories]].

## Examples

* Every [[cartesian monoidal category]] is necessarily symmetric monoidal, due to the essential uniqueness of the categorical [[product]]. This includes cases such as [[Set]], [[Cat]].

* For $k$ some [[field]], the category [[Vect]] of $k$-vector spaces carries the standard structure of a [[monoidal category]] coming from the [[tensor product]], over $k$, of vector spaces. The standard braiding that identifies $V \otimes W$ with $W \otimes V$ by mapping homogeneous elements $v \otimes w$ to $w \otimes v$ obviously makes [[Vect]] into a symmetric monoidal category.

* The category of $\mathbb{Z}_2$-[[graded vector space]]s, on the other hand, has two different symmetric monoidal extensions of the standard [[tensor product]] monoidal structure. One is the trivial one from above, the other is the one that induces a a sign when two odd-graded vectors $v$ and $w$ are passed past each other : $v \otimes w \mapsto - w \otimes v$. This non-trivial symmetric monoidal structure on $Vect[\mathbb{Z}_2]$ defines the symmetric monoidal category of [[super vector spaces]].

* The monoidal category of [[graded modules]] over a [[commutative ring]] (with the usual [[tensor product of modules|tensor product of graded modules]]) can be made into a braided monoidal category with the braiding

$$
  \array{
    V \otimes W &\longrightarrow& W \otimes V
    \\
    x \otimes y &\mapsto& y \otimes x
  }
  \,.
$$

The braiding $x \otimes y \mapsto (-1)^{|x| |y|} y \otimes x$ (where $|x|$ and $|y|$ denote the degrees) is also commonly used.

  More generally, for any invertible element $u$ of the base ring, there is the [[braided monoidal category|braiding]] $x \otimes y \mapsto u^{|x| |y|} y \otimes x$, and these braidings are the only possible. The resulting [[braided monoidal category]] is symmetric if and only if $u^2 = 1$.

## Related concepts

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

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

  * [[permutative category]], [[bipermutative category]]

  * [[symmetric monoidal dagger category]]

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

* [[symmetric monoidal closed category]]

* [[equivariant symmetric monoidal category]]

* [[symmetric bicategory]]

* [[symmetric monoidal groupoid]]

  * [[symmetric 2-group]], [[abelian ∞-group]]

* [[supply in a monoidal category]]

* [[commutative monoidal category]]

## References

Original references:

* {#MacLane} [[Saunders Mac Lane]], _Natural Associativity and Commutativity_ , Rice University Studies **49** (1963) 28-46 [[rice:1911/62865](https://scholarship.rice.edu/handle/1911/62865)]

* [[Jean Bénabou]], _Algèbre élémentaire dans les catégories_ (1964), C. R. Acad. Sci. Paris **258** (1964) 771-774 [[ark:12148/bpt6k40102/f817](https://gallica.bnf.fr/ark:/12148/bpt6k40102/f817)]

* {#EilenbergKelly65} [[Samuel Eilenberg]], [[G. Max Kelly]],  Part III of: *Closed Categories*, [[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)]

Textbook accounts:

* {#Borceux94} [[Francis Borceux]], Section 6.1 of: *Handbook of Categorical Algebra* Vol. 2: *Categories and Structures* $[$[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)$]$, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994)

* [[Saunders MacLane]], Ch. XI of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#EGNO15} [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *Tensor Categories*, AMS Mathematical Surveys and Monographs **205** (2015) $[$[ISBN:978-1-4704-3441-0](https://bookstore.ams.org/surv-205), [pdf](http://www-math.mit.edu/~etingof/egnobookfinal.pdf)$]$

  > (focused on [[tensor categories]])

Exposition of basics of [[monoidal categories]] and [[categorical algebra]]:

* _[[geometry of physics -- categories and toposes]]_, Section 2: _[Basic notions of categorical algebra](geometry+of+physics+--+categories+and+toposes#BasicNotionsOfCategoricalAlgebra)_

A survey of definitions of symmetric monoidal categories, [[symmetric monoidal functor]]s and [[symmetric monoidal natural transformation]]s, is also in

* [[John Baez]], _[Some definitions everyone should know](http://math.ucr.edu/home/baez/qg-fall2004/definitions.pdf)_

For an elementary introduction to symmetric monoidal categories using string diagrams, see:

* [[John Baez]] and [[Mike Stay]], [Physics, topology, logic and computation: a Rosetta Stone](http://math.ucr.edu/home/baez/rosetta.pdf).

The theorem that symmetric monoidal categories model all connective spectra is due to

* R. Thomason, _Symmetric monoidal categories model all connective spectra_ , Theory and applications of Categories, Vol. 1, No. 5, (1995) pp. 78-118
 {#Thomason}

More discussion is in

* [[Michael Mandell]], _An Inverse K-Theory Functor_ ([arXiv:1002.3622](http://arxiv.org/abs/1002.3622))
 {#Mandell}

* {#FongSpivak} Brendan Fong and David I, Spivak, *Supplying bells and whistles in symmetric monoidal categories*, [arxiv](https://arxiv.org/abs/1908.02633), 2019

* {#Schaeppi} Daniel Schäppi, *Ind-abelian categories and quasi-coherent sheaves*, [arxiv](https://arxiv.org/abs/1211.3678), 2014

* Amit Sharma, _Symmetric monoidal categories and $\Gamma$-categories_
Theory and applications of Categories, Vol. 35, No. 14, (2020) pp. 417-512
 {#Sharma}

* Stefano Kasangian and Fabio Rossi. _Some remarks on symmetry for a monoidal category_. Bulletin of the Australian Mathematical Society 23.2 (1981): 209-214.

* J. M. Egger. _On involutive monoidal categories_. Theory and Applications of Categories 25.14 (2011): 368-393.

[[!redirects symmetric monoidal categories]]
[[!redirects symmetric monoidal structure]]
[[!redirects symmetric monoidal structures]]

[[!redirects symmetric monoidal]]


symmetric monoidal closed category

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## Idea

A symmetric monoidal closed category is

* a [[monoidal category]]

which as such is also:

1. [[symmetric monoidal category|symmetric monoidal]]

1. [[closed monoidal category|closed monoidal]].

## Properties

### Relation between monoidal and strong monads

For [[monads]] on symmetric monoidal closed categories there is a close relation between [[structures]] of [[monoidal monads]] and of [[strong monads]].

For the moment see at *[enriched monad -- Relation to strong and monoidal monads](enriched}monad#RelationToStrongMonads)* for more on this.

## Examples

\begin{example}
  Any [[cartesian closed category]] is symmetric monoidal closed.
\end{example}

\begin{example}
 A *[[Bénabou cosmos]]* (a good [[base for enrichment]] in [[enriched category theory]]) is defined to be a [[bicomplete category|bicomplete]] symmetric monoidal closed category.
\end{example}

## Related concepts

* [[Bénabou cosmos]]

## References

### General

* {#EilenbergKelly65} [[Samuel Eilenberg]], [[G. Max Kelly]],  §III and specifically §III.6 in: *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)]

* {#Kelly82} [[Max Kelly]], §1.4, 1.5 in: _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories **10** (2005) 1-136  [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), [tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]

* {#Borceux94} [[Francis Borceux]], §6.1 of: *Categories and Structures*, Vol 2 of: _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)]

### Examples

Proof that the [[funny tensor product]] of [[categories]] is the only other [[symmetric closed monoidal category|symmetric closed monoidal structure]] on [[Cat]] besides the [[cartesian monoidal category|cartesian monoidal structure]]:

* {#FoltzLairKelly80} [[François Foltz]], [[Christian Lair]], [[G. M. Kelly]], *Algebraic categories with few monoidal biclosed structures or none*, Journal of Pure and Applied Algebra **17** 2 (1980) 171-177 [[pdf](https://core.ac.uk/download/pdf/82322397.pdf), doi:10.1016/0022-4049(80)90082-1]

[[!redirects symmetric monoidal closed categories]]

[[!redirects closed symmetric monoidal category]]
[[!redirects closed symmetric monoidal categories]]

[[!redirects closed symmetric monoidal structure]]
[[!redirects closed symmetric monoidal structures]]

[[!redirects symmetric closed monoidal category]]
[[!redirects symmetric closed monoidal categories]]


symmetric monoidal dagger category

[[!redirects symmetric monoidal dagger categories]]

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## Definition ##

A **symmetric monoidal dagger category** is a [[braided monoidal dagger category]] $C$ which is also a [[symmetric monoidal category]], i.e. such that for all [[objects]] $A \in Ob(C)$ and $B \in Ob(C)$, $\beta_{A,B} \circ \beta_{B, A} = \Iota_{A,B}$.

## Examples ##

* [[compact closed dagger category]]

* [[symmetric monoidal groupoid]], [[2-group]]

## See also ##

* [[dagger category]]

* [[monoidal dagger category]]

* [[braided monoidal dagger category]]

* [[symmetric monoidal category]]


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## Definition

A **symmetric monoidal $\dagger$-category** is a [[symmetric monoidal category]] that is also a $\dagger$-[[dagger-category|category]] for which:

1. $(f \otimes g)^\dagger = f^\dagger \otimes g^\dagger$ for every pair of [[morphisms]] $f,g$

1. the [[associator]], left and right [[unitor|unitors]], and [[braiding]] are all [[unitary morphism|unitary]].

If the category is also a [[compact closed category]] in a compatible way, then it is called a [[dagger-compact category]].

## References

* [[Peter Selinger|P. Selinger]], Dagger compact closed categories and completely positive maps, _Proceedings of the 3rd International Workshop on Quantum Programming Languages_, Chicago, June 30–July 1, 2005. [web](http://www.mscs.dal.ca/~selinger/papers.html#dagger)

[[!redirects symmetric monoidal †-category]]
[[!redirects symmetric monoidal †-categories]]
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symmetric monoidal functor

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## Idea

A _symmetric monoidal functor_ is a [[functor]] $F : C \to D$ between [[symmetric monoidal category|symmetric monoidal categories]] that is a [[monoidal functor]] which respects the symmetry on both sides.

## Definition

A (lax) [[monoidal functor]] $F : (C,\otimes) \to (D, \otimes)$, with monoidal structure $\nabla$, between [[symmetric monoidal categories]] is **symmetric** if for all $A,B \in C$ the diagram

$$
  \array{
    F A \otimes F B &\stackrel{\sigma}{\to}& F B \otimes F A
    \\
    {}^{\mathllap{\nabla_{A,B}}}\downarrow
    &&
    \downarrow^{\mathrlap{\nabla_{B,A}}}
    \\
    F(A\otimes B) &\stackrel{F(\sigma)}{\to}& F(B \otimes A)
  }
$$

commutes, where $\sigma$ denotes the symmetry isomorphism both of $C$ and $D$.

As long as it goes between symmetric monoidal categories a symmetric monoidal functor is the same as a [[braided monoidal functor]].

## Properties

+-- {: .num_prop #SymmetricMonoidalFunctorInducesFunctorOnCommutativeMonoids}
###### Proposition
**([[symmetric monoidal functor]] induces [[functor]] on [[commutative monoid in a symmetric monoidal category|commutative monoids]])**

A symmetric monoidal functor

$$
  \left(\mathcal{C}_1, \otimes_1, \tau_1\right)
  \longrightarrow
  \left(\mathcal{C}_2, \otimes_2, \tau_2\right)
$$

between two symmetric monoidal categories canonically preserves [[commutative monoid in a symmetric monoidal category|commutative monoids]] and extends to a functor between categories of commutative monoids (see [here](geometry+of+physics+--+categories+and+toposes#MonoidsPreservedByLaxMonoidalFunctor) for more)

$$
  CMon\left(\mathcal{C}_1, \otimes_1, \tau_1\right)
  \longrightarrow
  CMon\left(\mathcal{C}_2, \otimes_2, \tau_2\right)
$$

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## Examples

+-- {: .num_example #IdentityFunctorOnCategoryOfChainComplexesOfSuperVectorSpaces}
###### Example
**([[identity functor]] on [[category of chain complexes of super vector spaces]])**

The [[category of chain complexes of super vector spaces]] $Ch(Supervect)$ equipped with the [[tensor product of chain complexes]] carries two [[symmetric monoidal category|symmetric]] [[braidings]], $\tau_{Deligne}$ and $\tau_{Bernst}$ ([this Prop.](chain+complex+in+super+vector+spaces#SymmetricStructureOnCategoryOfChainComplexesOfSuperVectorSpaces)). The [[identity functor]] on $Ch(SuperVect)$ carries the structure of a [[strong monoidal functor|strong]] symmetric monoidal functor with respect to these two, making them equivalent. By Prop. \ref{SymmetricMonoidalFunctorInducesFunctorOnCommutativeMonoids} this in turn induces an [[equivalence of categories|equivalence]] on the catories of commutative monoids, which in this case are [[differential graded-commutative superalgebras]], with one of two equivalent grading conventions

$$
  dgcsAlg_{Deligne} \;\simeq\; dgcsAlg_{Bernstein}
$$

[[!include sign rules in homological superalgebra -- table]]

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## Related concepts

* [[monoidal functor]]

* [[strong monoidal functor]]

* [[lax monoidal functor]]

* [[oplax monoidal functor]]

* [[bilax monoidal functor]]

* [[Frobenius monoidal functor]]

* [[braided monoidal functor]]

* **symmetric monoidal functor**

## References

An exposition is in

* [[John Baez]], [Some definitions everyone should know](http://math.ucr.edu/home/baez/qg-fall2004/definitions.pdf).

[[!redirects symmetric monoidal functor]]
[[!redirects symmetric monoidal functors]]


symmetric monoidal groupoid

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## Idea

A *braided monoidal groupoid* is a [[braided monoidal category]] whose [[underlying]] [[category]] happens to be a [[groupoid]] (hence all whose [[morphisms]] are [[isomorphisms]]).

Equivalently: A [[braided monoidal groupoid]] whose dagger adjoint of the [[braiding]] is the opposite [[braiding]],

Equivalently, a [[1-type|1-truncated]] [[En-algebra|$E_3$-space]] or [[E-infinity algebra|$E_\infty$-space]].

## Definitions

A **symmetric monoidal groupoid** is a [[braided monoidal groupoid]] $G$ such that for all objects $A$ and $B$, the dagger adjoint or 2-sided inverse of the [[braiding]] $\beta_{A,B} : A \otimes B \cong^\dagger B \otimes A$ is the braiding $\beta_{B,A} : B \otimes A \cong^\dagger A \otimes B$

$$\beta_{A,B}^\dagger = \beta_{B,A}$$

## See also

* [[commutative monoid]]
* [[En-algebra]]
* [[groupoid]]
* [[monoidal groupoid]]
* [[braided monoidal groupoid]]
* [[k-tuply monoidal n-groupoid]]
* [[symmetric 2-group]]
* [[symmetric ring groupoid]]
* [[symmetric monoidal category]]

[[!redirects symmetric monoidal groupoid]]
[[!redirects symmetric monoidal groupoids]]


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## Idea

A _symmetric monoidal natural transformation_ is a [[natural transformation]] between [[symmetric monoidal functors]] that respects the symmetric monoidal structure.

In fact, a symmetric monoidal natural transformation is the same as a [[monoidal natural transformation]] (as long as it goes between symmetric monoidal functors).

## Related concepts

* [[braided monoidal natural transformation]]

## References

See, for example,

* [[John Baez]], _Some definitions everyone should know_, [pdf](http://math.ucr.edu/home/baez/qg-fall2004/definitions.pdf).


symmetric monoidal structure on spectrum objects

## Idea

Given a [[category]] $C$ one can consider the [[category]] of [[spectrum objects]] in $C$.  When $C$ is [[symmetric monoidal category|symmetric monoidal]], there is an induced [[symmetric monoidal structure]] on the [[category]] of [[spectrum objects]], generalizing the [[symmetric monoidal smash product of spectra]].

## Definitions

### Symmetric monoidal category of spectrum objects

Let $T$ be an [[object]] of a [[symmetric monoidal category]] $C$.  Following [Ayoub](#Ayoub) we can define a [[category]] $\Spect^\Sigma_T(C)$ of symmetric $T$-spectra in $C$ as described at [[spectrum object]].

In that case the category of $\Sigma$-[[symmetric sequences]] $\Seq(\Sigma, C)$ inherits a [[symmetric monoidal structure]] as described at [[symmetric sequence]].

Consider the [[symmetric sequence]] $Sym(T)$ whose $n$th component is $Sym(T)_n = T^{\otimes n}$ (with the [[actions]] of the [[symmetric group]] $\Sigma_n$ by permuting the $n$ factors).  It turns out that $Sym(T)$ is a commutative [[algebra object]] in the [[symmetric monoidal category]] $Seq(\Sigma, C)$.

As a consequence there is a [[symmetric monoidal structure]] on $Mod_\ell(Sym(T))$ where $X \otimes_{Sym(T)} Y$ is defined as the [[coequalizer]] of the diagram
  $$ X \otimes Sym(T) \otimes Y \rightrightarrows X \otimes Y. $$

By noting the tautological [[equivalence of categories]]
  $$ \Spect^\Sigma_T(C) \stackrel{\sim}{\longrightarrow} \Mod_\ell(Sym(T)) $$
one gets a [[symmetric monoidal structure]] on $Spect^\Sigma_T(C)$.

### Symmetric monoidal (infinity,1)-category of spectrum objects

...

## See also

* [[spectrum object]]
* [[symmetric sequence]]
* [[symmetric monoidal smash product of spectra]]

## References

The last chapter of

* [[Joseph Ayoub]], _Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, I_. Astérisque, Vol. 314 (2008). Société Mathématique de France. ([pdf](http://user.math.uzh.ch/ayoub/PDF-Files/THESE.PDF))


symmetric multicategory

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## Idea

A _symmetric multicategory_ is a [[multicategory]] equipped with an [[action]] of the [[symmetric group]] $S_n$ on the [[set]] of $n$-ary operations, for all $n$, such that composition respects this action.

Symmetric multicategories are equivalently called _coloured [[symmetric operads]]_ over [[Set]]. See there for more details.

## Properties

### Closed monoidal structure

With respect to the [[Boardman-Vogt tensor product]] (see there for details) symmetric multicategories form a [[closed monoidal category|closed]] [[symmetric monoidal category]].

### Thomason model structure

A full subcategory of based symmetric multicategories admit a Thomason-style model structure that is [[Quillen equivalent]] to [[connective spectra]], according to [Fuentes-Keuthan](#FuentesKeuthan2019).

### References

* {#FuentesKeuthan2019} [[Daniel Fuentes-Keuthan]], _Modeling connective spectra via multicategories_, [arXiv:1909.11148](https://arxiv.org/abs/1909.11148)

[[!redirects symmetric multicategories]]