symmetric operad

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

## Idea

The notion of _[[operad]]_ comes in two broad flavors (apart from the choice of enriching category): _symmetric operads_ and _[[planar operads]]_.

Roughly, a planar operad consists of $n$-ary operations for all $n \in \mathbb{N}$ equipped with a suitable notion of compositon, while a _symmetric operad_ in addition is equipped with a compatible [[action]] of the [[symmetric group]] $\Sigma_n$ on the set (or [[object]] in the enriching category) of $n$-ary operations. A [[homomorphism]] of symmetric operads is then a morphism of planar operads that in addition respects this action.

The extra "symmetry" structure carried by symmetric operads is crucial for the behaviour of the category of operads in many applications (see the examples below). Notice that it does not so much affect the idea of what a single operad is. In particular, symmetric operads are not restricted to encoding algebraic symmetric structures with symmetric $n$-ary operations! Rather, only the fixed points in the $n$-ary operations of the $\Sigma_n$-action are symmetric operations. If $\Sigma_n$ [[free action|acts freely]], then the corresponding $n$-ary operations are still maximally non-symmetric themselves.

The central example illustrating this are the operads [[Comm]] and [[Assoc]]. Regarded as symmetric [[Set]]-enriched operads, [[Comm]] has the singleton set in each degree, with _trivial_ $\Sigma_n$-action, while [[Assoc]] has $\Sigma_n$ in each degree, freely acting on itself.

Therefore [[Comm]] is the [[terminal object]] in the [[category]] of symmetric operads (while [[Assoc]], regarded as a [[planar operad]], is the terminal object in that category).

Multi-coured symmetric operads are equivalently known also as _[[symmetric multicategories]]_.

## Structures on the category of symmetric operads

### Boardman-Vogt tensor product

The category of symmetric operads becomes a [[closed monoidal category|closed]] [[symmetric monoidal category]] for the [[Boardman-Vogt tensor product]].

### Model structure

For $V$ a suitable [[monoidal model category]], the category of $V$-enriched symmetric operads carries a good [[model structure on operads]]. See there for more details.

## Properties

### Relation to categories

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

Every [[locally small category]] $C$ may be regarded as a coloured symmetric operation $j_!(C)$ over set, with the objects of $C$ and colours, and with only unary operation, these being the morphisms in the category

$$
  j_!(C)(c_1, \cdots, c_n ; c)
  =
  \left\{
    \array{
       C(c_1, c) & if \, n = 1
       \\
       \emptyset & otherwise
    }
  \right.
  \,.
$$

=--

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

This [[functor]] $j_! : Cat \to Operad$ has a [[right adjoint]] $j^* : Operad \to Cat$ which sends an operad to the underlying category obtained by discarding all $n$-ary operations for $n \neq 1$.

There is a [[natural isomorphism]] $j * j_! \simeq id$.

=--

By the discussion at _[[adjoint functor]]_ this exhibits a [[coreflective subcategory]]

$$
  Cat
   \stackrel{\overset{j_!}{\hookrightarrow}}{\underset{j^*}{\leftarrow}}
  Operad
  \,.
$$

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

Let $\eta$ denote the symmetric operad with a single colour and no non-identity operation. Then the [[slice category]] of [[Operad]] over $\eta$ is equivalent to [[Cat]]

$$
  Cat \simeq Operad_{/\eta}
  \,.
$$

Because a morphism of operads $P \to \eta$ can exists precisely if there are no operations of arity other than 1 in $P$.

Under this identification the fuctor $j_!$ is the canonical projection out of the slice category

$$
  j_! : Cat \stackrel{\simeq}{\to} Operad_{/\eta} \to Operad
  \,.
$$

=--

For more on this see at _[[dendroidal set]]_ the section _[The full diagram of relations](http://ncatlab.org/nlab/show/dendroidal%20set#FullDiagram)_.

### Relation to planar operads
 {#RelationToPlanarOperads}

There is an evident [[forgetful functor]]

$$
  U : SymmetricOperad \to PlanarOperad
$$

to the [[category]] of [[planar operads]],
which forgets the [[action]] of the [[symmetric groups]]. This functor has a [[left adjoint]]

$$
  Symm : PlanarOperad \to SymmetricOperad
  \,.
$$

The [[free construction]] freely adds symmetric group actions.

For instance as a [[planar operad]], [[Assoc]] is the [[terminal object]] (has the point in each degree). Its symmetrization $Symm(Assoc)$ is still the operad for associative [[monoids]], now regarded as a symmetric operad, where it has the underlying set of the [[symmetric group]] $\Sigma_n$ in degree $n$. This is no longer the terminal object in $SymmetricOperad$, which instead is [[Comm]].

## Examples

### In $Set$

We list some examples of [[Set]]-enriched symmetric operads.

* For every [[symmetric monoidal category]] $C$, there is naturally the symmetric [[endomorphism operad]] $End(C)$.

  This establishes a reflective (but non-full) inclusion

  $$
    End : SymmMonCat \to SymmOperad
  $$

  and makes precise the way in which a (symmetric) operad is a generalization of a (symmetric) monoidal category.

  For any other symmetric operad $P$, a morphism of symmetric operads

  $$
    P \to End(C)
  $$

  is precisely an [[algebra over an operad]] over $P$ in $C$.

* The operad [[Comm]] for [[commutative monoids]] is the [[terminal object]] in symmetric $V$-operads, for instance for $V = $ [[Set]], [[sSet]], [[Top]], etc.

  It has a single $n$-ary operation for all $n \in \mathbb{N}$, with the [[symmetric group]] necessarily acting trivially in each degree.

  A morphism of operads

  $$
    A : Comm \to End(Vect)
  $$

  is precisely a commutative and [[associative algebra]] structure on a [[vector space]].

* The operad [[Assoc]] for [[monoids]] is, as a symmetric operad, the one with a single colour that has precisely $n!$ many operations in degree $n$, with the [[symmetric group]] acting freely on these.

  This means that there is a single $n$-ary operation "up to a choice of ordering of its arguments".

  A morphism of operads

  $$
    Assoc \to End(Vect)
  $$

  is precisely an [[associative algebra]] on a vector space.

* For every _non-planar finite rooted [[tree]]_ there is a symmetric operad freely generated by it. For more on this see the section _[Trees and free operads](http://ncatlab.org/nlab/show/dendroidal+set#Trees)_ at _[[dendroidal set]]_.

### In $Top$

(...)

## Related concepts

* [[planar operad]]

* [[symmetric multicategory]]

## References

An original source is

* [[Peter May]], _The geometry of iterated loop spaces_, Lectures Notes in Mathematics, Vol. 271, Springer-Verlag, Berlin-New York, (1972).

A survey of the basic notions of symmetric operads is for instance in section 1 of

* [[Clemens Berger]], [[Ieke Moerdijk]], _Resolution of coloured operads and rectification of homotopy algebras_ Contemp. Math. 431 (2007) 31-58 ([arXiv:0512576](http://arxiv.org/abs/math/0512576))

See the references at _[[operad]]_ for more.

Expression of symmetric operads as [[polynomial functor|polynomial]] [[2-monads]] is discussed in

* [[Joachim Kock]], _Data types with symmetries and polynomial functors over groupoids_,  28th Conference on the Mathematical Foundations of Programming Semantics (Bath, June 2012); in Electronic Notes in Theoretical Computer Science.  ([arXiv:1210.0828](http://arxiv.org/abs/1210.0828))

* {#Weber14} [[Mark Weber]], _Operads as polynomial 2-monads_ ([arXiv:1412.7599](http://arxiv.org/abs/1412.7599))

[[!redirects symmetric operads]]


symmetric powers in a symmetric monoidal (Q plus)-linear category

[[!redirects symmetric powers in symmetric monoidal (Q plus)-linear categories]]

#Contents#
* table of contents
{:toc}

##Idea

A conjectural characterization of [[symmetric powers]] in symmetric monoidal $\mathbb{Q}^{+}$-[[linear category|linear categories]] is given on this page. The description in terms of [[universal properties]] would be equivalent to a much longer one but much more constructive in algebraic terms. It would be similar to the characterization of products known as [[Fox theorem]] but where [[products]] are replaced by [[symmetric powers]]. We work in symmetric monoidal categories enriched over modules over a $\mathbb{Q}^{+}$-algebra. The adaptation to [[CMon-enriched symmetric monoidal categories]] would involve symmetric and [[divided powers]] which are equal in this simpler setting. This conjecture seems to be true and if it is really the case, the proof is not trivial.

##Conjecture
We suppose that the symmetric monoidal categories are strict monoidal categories.

\begin{definition}
We say that a symmetric monoidal category $\mathcal{C}$ possesses the $n$-th symmetric power ($n \ge 0$) iff it possesses this coequalizer for every $A \in \mathcal{C}$:
\begin{tikzcd}
A^{\otimes n} \arrow[rr, "\sigma", shift left=3] \arrow[rr, "...", shift right=3] &  & A^{\otimes n} \arrow[rr, "r_{n}(A)"] &  & S_{n}(A)
\end{tikzcd}
and $r_{n}(A):A^{\otimes n} \rightarrow S_{n}(A)$ is a natural transformation.
\end{definition}

\begin{definition}
A functorial special bicommutative graded bimonoid in a $CMon$-enriched symmetric monoidal category is given by:

* a family $(S_{n}:\mathcal{C}\rightarrow \mathcal{C})_{n \ge 0}$ of [[endofunctors]]
* a family $(\nabla_{n,p}:S_{n}(A) \otimes S_{p}(A) \rightarrow S_{n+p}(A))_{n,p \ge 0}$ of [[natural transformations]]
* a family $(\Delta_{n,p}:S_{n+p}(A) \rightarrow S_{n}(A) \otimes S_{p}(A))_{n,p \ge 0}$ of natural transformations

such that:

* $S_{0}(A)=I$
* $S_{1}(A)=A$
* for every $A \in \mathcal{C}$, $(A,(\nabla_{n,p})_{n,p \ge 0}, (\Delta_{n,p})_{n,p \ge 0})$) is a special bicommutative [[graded bimonoid]] ie.:

* $\gamma;\nabla_{p,n} = \nabla_{n,p}$ (commutativity)
* $\Delta_{p,n};\gamma = \Delta_{n,p}$ (cocommutativity)
* $\nabla_{n,0}=1$ (unitality)
* $\Delta_{n,0}=1$ (counitality)
* $(\nabla_{n,p} \otimes 1);\nabla_{n+p,q} = (1 \otimes \nabla_{p,q});\nabla_{n,p+q}$ (associativity)
* $\Delta_{n+p,q};(\delta_{n,p} \otimes 1) = \Delta_{n+p,q};(1 \otimes \Delta_{p,q})$
(coassociativity)
* if $n+p = q+r$, then $\nabla_{n,p};\Delta_{q,r} = \underset{\substack{a,b,c,d \ge 0 \\ a+b=n \\ c+d=p \\ a+c=q \\ b+d=r}}{\sum} (\Delta_{a,b} \otimes \Delta_{c,d});(1 \otimes \gamma \otimes 1);(\nabla_{a,c} \otimes \nabla_{b,d})$
(compatibility multiplication/comultiplication)
* $\Delta_{n,p};\nabla_{n,p} = \binom{n+p}{n}$
(specialty)

\end{definition}

\begin{conjecture}
Let $R$ be a $\mathbb{Q}^{+}$-algebra. Let $\mathcal{C}$ be a symmetric monoidal category enriched over $R$-modules. Let $(S_{n}:\mathcal{C} \rightarrow \mathcal{C})_{n \ge 0}$ be a family of endofunctors such that $S_{0}(A)=I$ and $S_{1}(A) = A$. Then, there is a bijection between families of natural transformations $(r_{n}(A):A^{\otimes_{n}} \rightarrow S_{n}(A))_{n \ge 0}$ which make $(S_{n}(A))$ symmetric powers and families of natural transformations $(\nabla_{n,p}:S_{n}(A) \otimes S_{p}(A) \rightarrow S_{n+p}(A))_{n,p \ge 0}, (\Delta_{n,p}:S_{n+p}(A) \rightarrow S_{n}(A) \otimes S_{p}(A))_{n,p \ge 0}$ which make $(S_{n}(A))$ a functorial special bicommutative graded bimonoid.
\end{conjecture}

## Towards a proof

### $n^{th}$ symmetric power in symmetric monoidal categories enriched over modules over a $\mathbb{Q}^{+}$-algebra

\begin{proposition}
Let $R$ be a $\mathbb{Q}^{+}$-algebra and $\mathcal{C}$ be a symmetric monoidal category enriched over $R$-modules. Let $n \ge 0$ and $S_{n}:\mathcal{C} \rightarrow \mathcal{C}$ be an endofunctor. There is a bijection between natural transformations $r_{n}:A^{\otimes n} \rightarrow S_{n}(A)$ which make $(S_{n}(A))$ symmetric powers and pairs of natural transformations $(r_{n}:A^{\otimes n} \rightarrow S_{n}(A), s_{n}:S_{n}(A) \rightarrow A^{\otimes n})$ such that:

* $s_{n};r_{n}=1:S_{n}(A) \rightarrow S_{n}(A)$
* $r_{n};s_{n} = \frac{1}{n!} \underset{\sigma \in \mathfrak{S}_{n}}{\sum}\sigma:A^{\otimes n} \rightarrow A^{\otimes n}$

\end{proposition}
\begin{proof}
Suppose that we have a natural transformation $r_{n}:A^{\otimes n} \rightarrow S_{n}(A)$ which make $(S_{n}(A))$ symmetric powers. Define the natural transformation $e_{n}=\frac{1}{n!} \underset{\sigma \in \mathfrak{S}_{n}}{\sum}\sigma:A^{\otimes n} \rightarrow A^{\otimes}$. For every permutation $\rho \in \mathfrak{S}_{n}$, we have $\rho;e_{n}=\rho;\frac{1}{n!} \underset{\sigma \in \mathfrak{S}_{n}}{\sum}\sigma = \frac{1}{n!} \underset{\sigma \in \mathfrak{S}_{n}}{\sum}\rho;\sigma $$= \frac{1}{n!} \underset{\sigma \in \mathfrak{S}_{n}}{\sum} \sigma = e_{n}$ by using a change of variables in the last step. We thus have this factorization for every $A \in \mathcal{C}$:
\begin{tikzcd}
A^{\otimes n} \arrow[r, "r_{n}"] \arrow[rd, "e_{n}"'] & S_{n}(A) \arrow[d, "s_{n}"] \\
                                                      & A^{\otimes n}
\end{tikzcd}
It is not difficult to show that $s_{n}:S_{n}(A) \rightarrow A^{\otimes n}$ is a natural transformation.

Moreover $r_{n};(s_{n};r_{n}) = (r_{n};s_{n});r_{n} = e_{n};r_{n} = (\frac{1}{n!} \underset{\sigma \in \mathfrak{S}_{n}}{\sum}\sigma);r_{n}$ $ = \frac{1}{n!} \underset{\sigma \in \mathfrak{S}_{n}}{\sum}\sigma;r_{n} = \frac{1}{n!} \underset{\sigma \in \mathfrak{S}_{n}}{\sum}r_{n}=\frac{1}{n!}(n!r_{n})=r_{n}=r_{n};1$. We know that $r_{n}$ is an epimorphism and thus $s_{n};r_{n}=1$.

Suppose now that we have a pair of natural transformations $(r_{n}:A^{\otimes n} \rightarrow S_{n}(A), s_{n}:S_{n}(A) \rightarrow A^{\otimes n})$ such that:

* $s_{n};r_{n}=1:S_{n}(A) \rightarrow S_{n}(A)$
* $r_{n};s_{n} = \frac{1}{n!} \underset{\sigma \in \mathfrak{S}_{n}}{\sum}\sigma:A^{\otimes n} \rightarrow A^{\otimes n}$

Suppose that $f:A^{\otimes n} \rightarrow B$ is a morphism such that $\sigma;f=f$ for every $\sigma \in \mathfrak{S}_{n}$. Then $r_{n};(s_{n};f)=(r_{n};s_{n});f=(\frac{1}{n!} \underset{\sigma \in \mathfrak{S}_{n}}{\sum}\sigma);f = \frac{1}{n!} \underset{\sigma \in \mathfrak{S}_{n}}{\sum} \sigma;f = \frac{1}{n!} \underset{\sigma \in \mathfrak{S}_{n}}{\sum} f = \frac{1}{n!}(n!f)=f$. Suppose now that we have a morphism $g:S_{n}(A) \rightarrow B$ such that $f=r_{n};g$. Then, $s_{n};f=s_{n};(r_{n};g)=(s_{n};r_{n});g=g$. $s_{n};f$ is thus the only morphism $S_{n}(A) \rightarrow B$ such that $f=r_{n};(s_{n};f)$. Thus, $r_{n}:A^{\otimes n} \rightarrow S_{n}(A)$ makes $(S_{n}(A))$ symmetric powers.

\end{proof}

With this equivalent definition, we have already replaced the universal property defining symmetric powers by two natural transformations which verify some equations. The work is now to show the equivalence between these two equations and the ones defining a functorial special bicommutative graded bimonoid.

### From symmetric powers to creation/annihilation operators

\begin{definition}
Let $R$ be $\mathbb{Q}^{+}$-algebra and $\mathcal{C}$ a symmetric monoidal category enriched over $R$-modules. Suppose that we have a family $(S_{n}:\mathcal{C} \rightarrow \mathcal{C})_{n \ge 0}$ of endofunctors such that $S_{0}(A)=I$ and $S_{1}(A)=A$. Creation/annihilation operators are defined as two families $(c_{n}:S_{n}(A) \otimes A \rightarrow S_{n+1}(A))_{n \ge 0}$ and $(a_{n}:S_{n}(A) \rightarrow S_{n-1}(A) \otimes A)_{n \ge 1}$ such that:

[[n12.jpg:pic]]

and:

[[specialop2.jpg:pic]]

where $k$ must be read as $S_{k}(A)$. Be aware that "$n.$" depicts the multiplication by the scalar $n$.

\end{definition}

\begin{proposition}
Let $R$ be $\mathbb{Q}^{+}$-algebra and $\mathcal{C}$ a symmetric monoidal category enriched over $R$-modules. Suppose that we have a family $(S_{n}:\mathcal{C} \rightarrow \mathcal{C})_{0 \le n \le p}$ of endofunctors such that $S_{0}(A)=I$ and $S_{1}(A)=A$. Suppose that we have a family $(r_{n}:A^{\otimes n} \rightarrow S_{n}(A))_{0 \le n \le p}$ of natural transformations which make $(S_{n}A)_{0 \le n \le p}$ symmetric powers. We define creation/annihilation operators (for $0$ to $p$ particles) by putting:

\begin{tikzcd}
S_{n}(A) \otimes A \arrow[d, "s_{n} \otimes 1"'] \arrow[rd, "c_{n}"] &            &  & S_{n+1}(A) \arrow[d, "s_{n+1}"'] \arrow[rd, "a_{n}"] &                    \\
A^{\otimes (n+1)} \arrow[r, "r_{n+1}"']                              & S_{n+1}(A) &  & A^{\otimes (n+1)} \arrow[r, "r_{n}\otimes 1"']       & S_{n}(A) \otimes A
\end{tikzcd}

\end{proposition}

### From symmetric powers to functorial special bicommutative graded bimonoid

\begin{proposition}
Let $R$ be $\mathbb{Q}^{+}$-algebra and $\mathcal{C}$ a symmetric monoidal category enriched over $R$-modules. Suppose that we have a family $(S_{n}:\mathcal{C} \rightarrow \mathcal{C})_{0 \le n \le p}$ of endofunctors such that $S_{0}(A)=I$ and $S_{1}(A)=A$. Suppose that we have a family $(r_{n}:A^{\otimes n} \rightarrow S_{n}(A))_{0 \le n \le p}$ of natural transformations which make $(S_{n}A)_{0 \le n \le p}$ symmetric powers. We define a (bounded) functorial special bicommutative graded bimonoid by putting:

\begin{tikzcd}
S_{n}(A) \otimes S_{n'}(A) \arrow[d, "s_{n} \otimes s_{n'}"'] \arrow[rd, "{\nabla_{n,n'}}"] &            &  & S_{n+n'}(A) \arrow[d, "s_{n+n'}"'] \arrow[rd, "{\Delta_{n,n'}}"] &                           \\
A^{\otimes (n+n')} \arrow[r, "r_{n+n'}"']                                                  & S_{n+n'}(A) &  & A^{\otimes(n+n')} \arrow[r, "r_{n} \otimes r_{n'}"']            & S_{n}(A) \otimes S_{n'}(A)
\end{tikzcd}

for all $0 \le n,n' \le p$ such that $0 \le n+n' \le p$.

\end{proposition}

## Related concepts

[[graded codifferential category]]


symmetric product of circles

# Contents
* tic
{: toc}

## Idea

Recall that the $n$th [[symmetric product]] (or symmetric power) of a space, say $X$, is what you get if you take $n$ points in $X$ without considering the order in which they appear.  Since repetitions are allowed, one has to think carefully as to how to deal with multiplicities: either to respect them or to ignore them.
The first case, respecting multiplicities, means that $\{0,1,1\}$ and $\{0,0,1\}$ are not considered equivalent.  The second means that they are.

On this page we use the first definition, respecting multiplicities, and work with the [[circle]], $S^1$, as our original space.  The resulting spaces have straightforward descriptions and simple [[homotopy types]]: all are homotopic to the circle.  These descriptions can be found in [MR0210096](#MR0210096)

There is a link between this space and the [[unitary group]] of a finite dimensional complex [[Hilbert space]].  The conjugacy class of a [[unitary matrix]] is determined completely by its [[eigenvalue]]s with their multiplicities in that two unitary matrices are conjugate if and only if their sets-with-multiplicities ([[multiset|bags]]) of eigenvalues are the same.  The eigenvalues of a unitary matrix lie on a circle, but as we are considering the collections as sets and not ordered sets, we must consider the set of eigenvalues as a point in the symmetric product of the circle.  Hence knowing this space tells us about the space of conjugacy classes of unitary matrices.

## Definition

The notion of the *symmetric product* of a [[topological space]] is an application of the construction of the [[symmetric power]] in a symmetric monoidal category, where the category is $\Top$ and the monoidal structure is given by the cartesian product.

+-- {: .num_defn #sympcircle}
###### Definition
Let $X$ be a topological space.  Let $n \in \mathbb{N}$.  The **$n$th symmetric product** of $X$, written $SP^n(X)$, is quotient of $X^n$ by the action of the $n$th symmetric group acting by permuting the coordinates.
=--

On this page, we are most interested in the case $X = S^1$ but it will be instructive to consider also $\mathbb{C}^*$, the non-zero complex numbers, and $\mathbb{R}$.  This is because of the existence of the inclusion $S^1 \to \mathbb{C}^*$ and the exponential map $\mathbb{R} \to S^1$.

## Properties

The main property of $SP^n(S^1)$ is the following description, due to H. R. Morton in [MR0210096](#MR0210096).

+-- {: .num_prop #symcirdesc}
###### Proposition
The topological space $SP^n(S^1)$ is a fibre bundle over $S^1$ with fibre an $n-1$ [[simplex]].  If $n$ is odd, the bundle is orientable; if $n$ is even then it is not.

In particular, $SP^2(S^1)$ is a Möbius strip.
=--

Before proving this, let us give the descriptions of $SP^n(\mathbb{C}^*)$ and $SP^n(\mathbb{R})$.

+-- {: .num_prop #symcplx}
###### Proposition
The topological space $SP^n(\mathbb{C}^*)$ is homeomorphic to $\mathbb{C}^{(n-1)} \times \mathbb{C}^*$.
=--

+-- {: .proof}
###### Proof
The proof rests on giving an alternative description of $SP^n(\mathbb{C}^*)$ in which the lack of ordering of the points is built in at the start.  This description is given by taking the polynomial whose roots are the points (here we can see why it is important to take note of the multiplicities).

We define a map $SP^n(\mathbb{C}^*) \to \Poly_n$, the space of polynomials of degree at most $n$, by sending $[\zeta_1,\dots,\zeta_n]$ to the monic polynomial $(z - \zeta_1)\cdots (z - \zeta_n)$.  This is a [[homeomorphism]] onto its image.  That image is the space of monic polynomials with non-zero constant term (as all the $\zeta_j$ are non-zero).  There is an obvious homeomorphism of this space to $\mathbb{C}^{n-1} \times \mathbb{C}^*$ given by taking coefficients.
=--

This space is relevant because the construction $SP^n(-)$ respects homotopy equivalences.  Hence $SP^n(S^1)$ is homotopy equivalent to $S^1$ and the map $SP^n(S^1) \to S^1$ is given by multiplying the $n$ elements of a point in $SP^n(S^1)$.

+-- {: .num_cor #sphaus}
###### Corollary
The space $SP^n(S^1)$ is Hausdorff.
=--

+-- {: .proof}
###### Proof
It embeds in the Hausdorff space $SP^n(\mathbb{C}^*)$.
=--

+-- {: .num_prop #symreal}
###### Proposition
The space $SP^n(\mathbb{R})$ is homeomorphic to $\mathbb{R} \times (\mathbb{R}_+)^{n-1}$.
=--

+-- {: .proof}
###### Proof
The idea of this proof is that although in $SP^n(\mathbb{R})$ an $n$-tuple of points has no intrinsic order (due to the action of the symmetric group), we can always impose an order from "outside" by taking the points in the order in which they occur in $\mathbb{R}$.  Thus any element of $SP^n(\mathbb{R})$ has a unique representation as $(t_1,t_2,\dots,t_n)$ where $t_j \le t_{j+1}$.  We define the map $SP^n(\mathbb{R}) \to \mathbb{R} \times (\mathbb{R}_+)^{n-1}$ by:
$$
(t_1,t_2, \dots, t_n) \mapsto (t_1, t_2 - t_1, t_3 - t_2, \dots, t_n - t_{n-1}).
$$

There is another map which is worth mentioning.  The space $\mathbb{R}^n$ has an invariant subspace under the action of the symmetric group (the $n$-fold diagonal) and it is possible to find a homeomorphism $SP^n(\mathbb{R}) \to \mathbb{R} \times (\mathbb{R}_+)^{n-1}$ where the first map is the orthogonal projection onto this subspace.  The projection is given by the map which sends $\{t_1,\dots,t_n\}$ to the average value.  The fibre of this map is the quotient by the action of the symmetric group of the space of vectors in $\mathbb{R}^n$ whose components sum to zero.  As they sum to zero, knowing the differences between the points (in the order imposed from $\mathbb{R}$) is enough to specify the points themselves.  The resulting map is thus:
$$
(t_1,t_2,\dots,t_n) \mapsto \left(\frac{1}{n} \sum t_j, t_2 - t_1, \dots, t_n - t_{n-1}\right).
$$
=--

+-- {: .proof}
###### Proof of Proposition \ref{symcirdesc} ######

Now let us prove the proposition.  Guided by the description for $SP^n(\mathbb{C}^*)$, we define $p \colon SP^n(S^1) \to S^1$ by $p(\{\zeta_1,\dots,\zeta_n\}) = \zeta_1 \dots \zeta_n$.  Let us show that this is a fibre bundle.  For $\lambda \in S^1$ we define a map $p^{-1}(\lambda) \times (S^1 \setminus \{\overline{\lambda}\}) \to p^{-1}(S^1 \setminus \{\overline{\lambda}\})$ by:

$$
p(\{\zeta_1,\dots,\zeta_n\},\lambda e^{i t}) = \{\zeta_1 e^{i t/n}, \dots, \zeta_n e^{i t/n}\}
$$

where $t \in (-\pi,\pi)$.

It remains to determine the fibre at $1$ and the gluing map at, say, $-1$.

To do this, we consider the exponential map $SP^n(\mathbb{R}) \to SP^n(S^1)$.  We use the second description of $SP^n(\mathbb{R})$ as $\mathbb{R} \times (\mathbb{R}_+)^{n-1})$ to view $SP^n(\mathbb{R})$ as a bundle over $\mathbb{R}$.  With this picture, the exponential map $SP^n(\mathbb{R}) \to SP^n(S^1)$ is a bundle map covering the exponential map $\mathbb{R} \to S^1$.  In particular, we have an induced map $(\mathbb{R}_+)^{n-1} \to p^{-1}(1)$ given by the following recipe:

1. Start with $(r_1,r_2,\dots,r_{n-1}) \in (\mathbb{R}_+)^{n-1}$.
2. Form the sequence of partial sums $(s_1,s_2,\dots,s_n)$ where $s_k = \sum_{j \lt k} r_j$.  Note that $s_1 = 0$.
3. Adjust so that their average (equivalently, sum) is $0$ by setting $t_k = s_k - \frac{1}{n} \sum s_j$.
4. Apply the exponential function to produce $\{e^{2 \pi i t_k}\}$ in $SP^n(S^1)$.

To use this to identify the fibre of $p^{-1}(1)$ we restrict this to a simplex.  The simplex that we choose at the start is:
$$
\Delta^n = \{(r_1,r_2,\dots,r_{n-1}) \in [0,1]^n : \sum r_j \le 1\}.
$$
At the second step, this becomes:
$$
\{(s_1,s_2,\dots,s_n) \in [0,1]^n : s_j \le s_{j+1}\}.
$$
At the third step, this becomes:
$$
\{(t_1,t_2,\dots,t_n) \in \mathbb{R} : \sum t_j = 0,\; t_n \le t_1 + 1\}.
$$
Thus far, all the maps are by homeomorphisms of their ambient spaces and so are homeomorphisms as they are bijections.

We now wish to determine the image of this in $SP^n(S^1)$.  Let $\{\zeta_1,\dots,\zeta_n\}$ be a point in $p^{-1}(1)$.  We order them cyclically around the circle going anticlockwise.  To turn this into a proper order, we need to choose a first point.  If that point is a multiple point, we do not assume that the other points follow immediately after it: some may but the others may occur at the end of the list when we have gone all around the circle.  With a multiple point then the decision that that contains the first point also contains a partition of the multiplicities into "at the start of the list" and "at the end of the list".

Having chosen the first point, we now choose a preimage of that point in $\mathbb{R}$, say $t_1$.  This defines a logarithm on the circle with branch cut at the first point.  We then proceed around the circle, lifting the other points to $\mathbb{R}$ using the same logarithm.  If the chosen point had multiplicities and some are at the end of the list, then these are lifted consistently with having gone around the circle, which means that they are lifted to $t_1 + 1$.  The result of this lifting is a set of points $(t_1,t_2,\dots,t_n)$ with $t_j \le t_{j+1}$ and $t_n \le t_1 + 1$.  The sum of these points in the line is in the preimage of the product of the points in the circle and thus is an integer.

To see that we can obtain any integer for the sum, consider what happens if we change our choice of first point to the next point anticlockwise.  If our original point was multiple and there are others at the start of the list then this means that we shift one point to the end of the list.  If we have moved to a new point and it is multiple, then we put all of its points at the start of the list and none at the end.  When looking at what happens to the logarithm, we have shifted the cut around (possibly) and now lift starting at $t_2$.  The last point that we reach is the one below $t_1$, but we have gone around the circle so we actually reach $t_1 + 1$.  Thus the lift of the $n$-tuple of points is now $(t_2,t_3,\dots,t_n,t_1+1)$.  The sum has increased by $1$.  Hence by choosing the right first point, and the right start to the logarithm, we can ensure that we get an $n$-tuple of points in $\mathbb{R}$ which sum to $0$.  The conclusion of all of this is that at the fourth step, when restricting everything to the simplex, the map is surjective onto $p^{-1}(1)$.

It is also injective.  The above discussion shows that changing the choice of first point changes the sum by integral values and it is also clear that changing the starting point of the logarithm changes it by multiples of $n$.  It is therefore not possible to find two distinct starting points and two distinct lifts which both sum to $0$.

As the simplex is compact and $p^{-1}(1)$ is Hausdorff, the map $\Delta^n \to p^{-1}(1)$ is thus a homeomorphism.

The last step is to identify the transition map.  As we go around the circle, we need to adjust the above to take into account the variation in sum.  This can be done at the last step where we map to $SP^n(S^1)$ by multiplying each term by $e^{2 \pi i \tau/n}$, where $e^{2 \pi i \tau}$ is the desired product of all the points.  The effect that this has on the logarithm is, after a complete revolution, to add $\frac{1}{n}$ to each component.  This results in an $n$-tuple with sum $1$ so to correct that we have to move one term from the end to the beginning.  This results in a coordinate transformation $(t_1,t_2,\dots,t_n) \mapsto (t_n - 1, t_1, t_2, \dots, t_{n-1})$ which is orientable if $n$ is odd and unorientable if $n$ is even.
=--

Let us illustrate this for the case $n = 2$.  Then $SP^2(S^1)$ consists of equivalence classes of pairs of points $\{\zeta_1,\zeta_2\}$ in $S^1$, with repetition allowed.  In the fibre of $p$ at $1$ we have the condition $\zeta_1 \zeta_2 = 1$ and so we can write our pair as $\{\zeta,\overline{\zeta}\}$.

On the subset $\zeta \ne -1$ there is a clear parametrisation with domain $[0,1)$ given by $t \mapsto \{e^{i \pi t}, e^{-i \pi t}\}$.  The claim is that this extends to a homeomorphism $[0,1] \to p^{-1}(1)$ by sending $1$ to $\{-1,-1\}$.  We can define the inverse map directly by sending $\zeta$ to $\Re(\zeta)$ and then composing with an appropriate inverse trigonometric function, but it is more instructive to consider why this works.

Let us imagine that we approach $\{-1,-1\}$ and pass through it.  Both before and after, we have two dots moving around the circle mirroring each other.  At the crucial time, they meet at $-1$, and then move apart again.  Since they are mirroring each other, one comes back along the top part of the circle and one along the bottom.  The crucial piece of the puzzle is that we cannot tell which is which.  So as far as we are concerned, the two dots may have passed through each other or may have bounced back off each other again.

Hence the fibre at $p^{-1}(1)$ is $[0,1]$.

Now consider what happens as we move around the circle.  Suppose that we look at a path on the boundary which starts at the image of $0$ as we move around, then when we are at $e^{2 \pi i t} \in S^1$, we have two points at $e^{\pi i t}$.  After going round a full revolution, our double point has only reached $e^{\pi i} = -1$ which is the image of $1$.  Hence we have a Möbius strip.

## Conjugacy Classes of Unitary Matrices

The symmetric product $SP^n(S^1)$ can be identified with the space of conjugacy classes of unitary transformations on $\mathbb{C}^n$, aka unitary $n \times n$-matrices.  This is because the conjugacy class of a unitary transformation is characterised by the eigenvalues of a matrix in the class, including their multiplicities, and these lie in $S^1$.  Hence $U_n/conj \simeq SP^n(S^1)$ and the induced map $U_n \to S^1$ is the [[determinant]] mapping.

## References

* Morton, H. R. (1967). Symmetric products of the circle. _Proc. Cambridge Philos. Soc._, _63_, 349–352.  [MR0210096](http://www.ams.org/mathscinet-getitem?mr=210096)
{#MR0210096}


symmetric relation

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#### Relations
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=--
#### Graph theory
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[[!include graph theory - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}

## Definition

A (binary) [[relation]] $\sim$ on a set $A$ is __symmetric__ if any two elements that are related in one order are also related in the other order:
$$\forall (x, y: A),\; x \sim y \;\Rightarrow\; y \sim x$$

In the language of the $2$-poset-with-duals [[Rel]] of sets and relations, a relation $R: A \to A$ is __symmetric__ if it is contained in its reverse:
$$R \subseteq R^{op}$$
In that case, this containment is in fact an equality.

## Relation to graphs

A set with a symmetric relation is the same as a [[loop digraph object|loop]] [[digraph]] $(V, E, s:E \to V, t:E \to V)$ with a function $sym:E \to E$ such that

* for every $f \in E$, $s(f) =_V t(sym(f))$
* for every $f \in E$, $t(f) =_V s(sym(f))$
* for every $f \in E$, $sym(sym(f)) =_E f$

## Related concepts

* [[relation]]

  * [[reflexive relation]]

  * **symmetric relation**

* [[loop graph object]]

* [[equivalence relation]]


symmetric ring

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

## Idea

A [[free object|free]] [[commutative monoid]] object in [[Ab]]

## Definition

Given an [[abelian group]] $G$, the **symmetric ring** is a [[commutative ring]] $S(G)$ with an abelian group homomorphism $g:G \to S(G)$, such that for every other commutative ring $R$ with abelian group homomorphism $h:G \to R$, there is a unique commutative ring homomorphism $i:S(G) \to R$ such that $i \circ g = h$.

## See also

* [[commutative ring]]

* [[tensor ring]]

* [[symmetric algebra]]

[[!redirects symmetric rings]]


symmetric ring groupoid

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###Context###
#### Algebra
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[[!include higher algebra - contents]]
=--
#### Type theory
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[[!include type theory - contents]]
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#### Categorification
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[[!include categorification - contents]]
=--
#### Category theory
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[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A *symmetric ring groupoid* is a [[1-type|1-truncated]] [[E-infinity-ring|$E_\infty$-ring]].

Equivalently: A [[groupoid]] $G$ that is both a [[symmetric 2-group]] $(G, 0, \oplus)$ and a [[symmetric monoidal groupoid]] $(G, 1, \otimes)$ such that $\otimes$ [[distributivity|distributes]] over $\oplus$ satisfying certain higher [[coherence laws]] (given in [Kelly74](#Kelly74)).

## Properties

* The [[discrete groupoid]] of [[integers]] $\mathbb{Z}$ is the [[initial object|initial]] symmetric ring groupoid.

* The [[contractible]] groupoid or the [[trivial ring|trivial ring groupoid]] $0$ is the [[terminal object|terminal]] symmetric ring groupoid.

## See also

* [[commutative ring]]

* [[E-infinity-ring]]

* [[ring groupoid]]

* [[symmetric 2-group]]

* [[symmetric monoidal groupoid]]

## References

* {#Kelly74} [[G. M. Kelly]], _Coherence theorems for lax algebras and distributive laws_, Lecture Notes in Mathematics **420**, Springer Verlag, Berlin(1974) 281-375 $[$[doi:10.1007/BFb0063106](https://doi.org/10.1007/BFb0063106)$]$

[[!redirects symmetric ring groupoids]]


symmetric ring spectrum

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#### Stable Homotopy theory
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#Contents#
* table of contents
{:toc}

## Idea

A _symmetric ring spectrum_ is a [[ring spectrum]] modeled as a [[symmetric spectrum]]. Equivalently this is a [[monoid object]] with respect to the [[symmetric monoidal smash product of spectra]], for symmetric spectra.
Morever, if we regard symmetric spectra as $\mathbb{S}_{sym}$-[[modules]], as discussed at _[[Model categories of diagram spectra]]_, then this, in turn, is equivalent to a $\mathbb{S}_{sym}$-algebra, where $\mathbb{S}_{sym}$ is the standard model of the [[sphere spectrum]] as a symmetric spectrum.

There is a [[model structure for ring spectra|model structure for symmetric ring spectra]] ([MMSS 00](#MandellMaySchwedeShipley01)) under which symmetric ring spectra represent genuine [[A-infinity rings]], and commutative symmetric ring spectra represent genuine [[E-infinity rings]]. This fact is one of the key motivation for passing from [[sequential spectra]] to the richer model of [[symmetric spectra]] (and possibly further to other [[highly structured spectra]] such as [[orthogonal spectra]] and [[excisive functors]]).

Despite all this, the component expression of the structure in a symmetric ring spectrum, in the fashion of _[[functors with smash product]]_, is rather straightforward, and directly analogous to the structure in a [[dg-algebra]]: essentially there is for all pairs of indices $n_1, n_2$ a pairing between the component spaces of the spectrum in these degrees

$$
  E_{n_1}\wedge E_{n_2}\longrightarrow E_{n_1 + n_2}
$$

such that this respects the given action of the [[symmetric groups]] and of [[suspensions]], and such that it is is [[associativity|associative]] and [[unitality|unital]] in the evident way.

## Definition

Write $Top_{cg}^{\ast}$ for the [[category]] of [[pointed topological space|pointed]] [[compactly generated topological spaces]].

One minimal way to state the definition is as follows. We follow conventions as used at _[[model structure on orthogonal spectra]]_.

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

A **commutative symmetric ring spectrum** $E$ is

1. a sequence of component spaces $E_n \in Top^{\ast/}_{cg}$ for $n \in \mathbb{N}$;

1. a basepoint preserving continuous left [[action]] of the [[symmetric group]] $\Sigma(n)$ on $E_n$;

1. for all $n_1,n_2\in \mathbb{N}$ a _multiplication map_

   $$
     \mu_{n_1,n_2}
       \;\colon\;
      E_{n_1} \wedge E_{n_2} \longrightarrow E_{n_1 + n_2}
   $$

   (a morphism in $Top^{\ast/}_{cg}$)

1. two _unit maps_

   $$
     \iota_0 \;\colon\; S^0 \longrightarrow E_0
   $$

   $$
     \iota_1 \;\colon\; S^1 \longrightarrow E_1
   $$

such that

1. $\mu_{n_1,n_2}$ [[intertwiner|intertwines]] the $\Sigma(n_1) \times \Sigma(n_2)$-action;

1. (associativity) for all $n_1, n_2, n_3 \in \mathbb{N}$ the following [[commuting diagram|diagram commutes]] (where we are notationally suppressing the [[associators]] of $(Top^{\ast/}_{cg}, \wedge, S^0)$)

   $$
     \array{
       E_{n_1} \wedge E_{n_2} \wedge E_{n_3}
         &\overset{id \wedge \mu_{n_2,n_3}}{\longrightarrow}&
       E_{n_1} \wedge E_{n_2 + n_3}
       \\
       {}^{\mathllap{\mu_{n_1,n_2}\wedge id }}\downarrow
         &&
       \downarrow^{\mathrlap{\mu_{n_1, n_2 + n_3}}}
       \\
       E_{n_1 + n_2} \wedge E_{n_3}
         &\underset{\mu_{n_1 + n_2, n_3}}{\longrightarrow}&
       E_{n_1 + n_2 + n_3}
     }
     \,;
   $$

1. (unitality) for all $n \in \mathbb{N}$ the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       S^0 \wedge E_n
         &\overset{\iota_0 \wedge id}{\longrightarrow}&
       E_0 \wedge E_n
       \\
       &{}_{\mathllap{\ell^{Top^{\ast/}_{cg}}_{E_n}}}\searrow& \downarrow^{\mathrlap{\mu_{0,n}}}
       \\
       && E_n
     }
   $$

   and

   $$
     \array{
       E_n \wedge S^0
         &\overset{id \wedge \iota_0 }{\longrightarrow}&
       E_n \wedge E_0
       \\
       &{}_{\mathllap{r^{Top^{\ast/}_{cg}}_{E_n}}}\searrow& \downarrow^{\mathrlap{\mu_{n,0}}}
       \\
       && E_n
     }
   $$

1. (commutativity) for all $n_1, n_2 \in \mathbb{N}$ the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       E_{n_1} \wedge E_{n_2}
         &\overset{\tau^{Top^{\ast/}_{cg}}_{E_{n_1}, E_{n_2}}}{\longrightarrow}&
       E_{n_2} \wedge E_{n_1}
       \\
       {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow
         &&
       \downarrow^{\mathrlap{\mu_{n_2,n_1}}}
       \\
       E_{n_1 + n_2}
         &\underset{\chi_{n_1,n_2}}{\longrightarrow}&
       E_{n_2 + n_1}
     }
     \,,
   $$

where $\chi_{n_1,n_2} \in \Sigma(n_1 + n_2)$ denotes the [[permutation]] which [[shuffle|shuffles]] the first $n_1$ elements past the last $n_2$ elements.

A [[homomorphism]] of symmetric commutative ring spectra $f \colon E \longrightarrow E'$ is a sequence $f_n \;\colon\; E_n \longrightarrow E'_n$ of $\Sigma(n)$-equivariant pointed continuous functions such that the following [[commuting diagram|diagrams commute]] for all $n_1, n_2 \in \mathbb{N}$

$$
  \array{
    E_{n_1} \wedge E_{n_2}
     &\overset{f_{n_1} \wedge f_{n_2}}{\longrightarrow}&
    E'_{n_1} \wedge E'_{n_2}
    \\
    {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow
      &&
    \downarrow^{\mu_{n_2,n_1}}
    \\
    E_{n_1 + n_2}
     &\underset{\chi_{n_1, n_2}}{\longrightarrow}&
    E_{n_2 + n_1}
  }
$$

and $f_0 \circ \iota_0 = \iota_0$ and $f_1\circ \iota_1 = \iota_1$.

Write

$$
  CRing(SymSpec(Top_{cg}))
$$

for the resulting [[category]] of symmetric [[commutative ring spectra]].

There is an analogous definition of **[[orthogonal ring spectrum]]** and we write

$$
  CRing(OrthSpec(Top_{cg}))
$$

for the category that these form.

=--

([Schwede 12, def. 1.3](#Schwede12))

## Related concepts

* [[orthogonal ring spectrum]]

## References

* {#MandellMaySchwedeShipley01} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], _[[Model categories of diagram spectra]]_, Proceedings of the London Mathematical Society, 82 (2001), 441-512 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf))

* {#Schwede12} [[Stefan Schwede]], _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

The structure of a [[symmetric ring spectrum]] on [[KO]] is discussed in

* {#Joachim01} [[Michael Joachim]], _A symmetric ring spectrum representing
$KO$-theory_, Topology 40 (2001) 299-308

[[!redirects symmetric ring spectra]]


symmetric sequence

#Contents#
* table of contents
{:toc}

## Idea

A symmetric sequence is a [[sequence]] of [[objects]] where the $n$th [[object]] has an [[action]] of the $n$th [[symmetric group]].

A symmetric sequence is a [[species]] by another name. Meaning: they are categorically equivalent notions.

## Definition

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

1.  Let $C$ be a [[category]] and $G$ a [[group]].  A **$G$-representation** of $C$ is a functor $\bullet\{G\} \to C$, where $\bullet\{G\}$ is the [[category]] with a single object $\bullet$ and $\Hom(\bullet, \bullet) = G$.  Explicitly, a $G$-representation of $C$ is the data of an object $X \in C$ together with an [[group action|action]] $a : G \to \Aut_C(X)$.  We write $Rep(G, C)$ for the category of $G$-representations of $C$.

2.  Let $\Phi$ be a [[graded monoid]] in the category of [[groups]].  Explicitly this is the data of [[groups]] $\Phi_n$ for all $n \in \mathbf{N}$ with morphisms $\Phi_m \times \Phi_n \to \Phi_{m+n}$ for all $m,n \ge 0$ (subject to various axioms...).  $\Phi$ is usually either $\Sigma = (\Sigma_n)_n$, the [[graded monoid]] of [[symmetric groups]], or $1 = (1_n)_n$, the [[graded monoid]] of [[trivial group]]s.

3.  A **$\Phi$-symmetric sequence** in $C$ is a [[sequence]] of $\Phi_n$-representations for $n \ge 0$:
  $$ Seq(\Phi, C) = \sqcup_{n \ge 0} Rep(\Phi_n, C) $$
In other words a $\Phi$-symmetric sequence is a sequence of objects $(X_n)_{n \ge 0}$ together with [[actions]] $a_n : \Phi_n \to \Aut_C(X_n)$.  When $\Phi = \Sigma$, the [[graded monoid]] of [[symmetric groups]], we say simply "symmetric sequence".

=--

In the case that the graded group of interest is indeed $\Sigma$, we can define a $\Sigma$-symmetric sequence somewhat more simply:

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

A **$\Sigma$-symmetric sequence** in a [[symmetric monoidal category]] $C$ is a functor from $FinSet$, the [[category]] of [[finite sets]] and [[bijections]], to $C$.

=--

The relationship between the two definitions is that given a functor $F:FinSet\to C$, we have a sequence of objects for each $n$ associated to the finite set with $n$ elements. The action of $\Sigma_n$ on these objects comes from the fact that for every [[permutation]] in $\Sigma_n$ there is an associated morphism in $FinSet$. Sometimes the category $FinSet$ is replaced with its [[skeleton]], the category of all finite [[ordinals]] with all bijections between them. This latter category is sometimes denoted $FinOrd$ or just $\Sigma$. In the latter case, we sometimes call $\Sigma$-symmetric sequences just "$\Sigma$-sequences."

## Symmetric monoidal structure

4.  Let $\alpha : H \to G$ be a [[homomorphism]] of [[groups]] and consider the [[restriction of scalars]] functor
  $$ \Res^G_H : \Rep(G, C) \to \Rep(H, C) $$
which is defined in the obvious way.  It admits a left adjoint
  $$ Ind^G_H : \Rep(H, C) \to \Rep(G, C) $$
called the _induced representation_ functor.

5.  Suppose now that $C$ has a [[symmetric monoidal structure]].  Assume also that $C$ admits [[coproducts]] and that the functors $X \otimes -$ commute with them (for all $X \in C$).  Then there is an induced [[symmetric monoidal structure]] on $Seq(\Phi, C)$.  Given symmetric sequences $X = (X_n)_n$ and $Y = (Y_n)_n$, we define $X \otimes Y$ as the symmetric sequence which has in the $n$th component
  $$ (X \otimes Y)_n = \sqcup_{p+q=n} \Ind^{\Phi_n}_{\Phi_p \times \Phi_q} (X_p \otimes Y_q) $$
where $\Phi_p \times \Phi_q \to \Phi_n$ is the canonical morphism which is part of the [[structure]] of the [[graded monoid]] $\Phi = (\Phi_n)_n$.  The [[unit]] with respect to this [[monoidal structure]] is given by $1 = (1, \emptyset, \emptyset, ...)$.

## Relationship to Operads

Symmetric sequences are useful in defining [[operads]] ([[symmetric operads]]) in [[symmetric monoidal categories]]. In particular, an operad in a symmetric monoidal category $C$ can be defined to be a [[monoid]] in the category of symmetric sequences of $C$. See, for instance, Definition 2.2.9 of [Ching](#Ching), and see at _[operad -- Definition as monoid](operad#DefinitionAsMonoid)_.

## See also

* [[spectra]], [[symmetric spectra]]

* [[colored symmetric sequence]]

## 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))

* {#Ching} [[Michael Ching]], _Bar constructions for topological operads and the Goodwillie derivatives of the identity_ (PhD Thesis), [Available Here](https://dspace.mit.edu/bitstream/handle/1721.1/27881/61212201-MIT.pdf?sequence=2).

[[!redirects symmetric sequences]]
[[!redirects sigma sequence]]
[[!redirects sigma sequences]]


symmetric set

#Contents#
* table of contents
{:toc}

## Idea

A __symmetric set__ (or __symmetric simplicial set__) is a [[simplicial set]], $X$, equipped with additional _transposition maps_ $t^n_i: X_n \to X_n$ for $i=0,\ldots,n-1$.  These transition maps generate an [[action]] of the [[symmetric group]] $\Sigma_{n+1}$ on $X_n$ and satisfy certain commutation relations with the face and degeneracy maps.  The result is that a symmetric set is a [[presheaf]] of sets on the category [[FinSet]]$_+$ of nonempty [[finite set]]s (or on its [[skeleton]]).

An analogy to keep in mind is
: symmetric set : simplicial set :: [[groupoid]] : [[category]].

This analogy can be formalized by noticing that the skeletal category of finite sets is simply the [[full subcategory]] of [[Cat]] whose objects are the [[localization]]s $[n]^{-1}[n]$ which are groupoids. By the universal property of localization, the usual (simplicial) [[nerve]] of a groupoid has a canonical symmetric structure.

Grandis proves that the fundamental groupoid functor $!Smp \to Gpd$ from symmetric sets to groupoids is left adjoint to a natural functor $Gpd \to !Smp$, the symmetric nerve of a groupoid, and preserves all colimits - a [[van Kampen theorem]]. Similar results hold in all higher dimensions.

The notion of [[cyclic set]] is intermediate between symmetric sets and simplicial sets.  In particular, any symmetric set, such as the nerve of a groupoid, also has a cyclic structure.

## Related concepts

* [[symmetric comonad]]

* [[cyclic set]],  [[skew-simplicial set]]

## References

* {#Lawvere21}[[William Lawvere]], _Toposes generated by codiscrete objects in combinatorial topology and functional analysis_, Reprints in Theory and Applications of Categories, No. 27 (2021) pp. 1-11, [pdf](http://tac.mta.ca/tac/reprints/articles/27/tr27abs.html).

* [[Marco Grandis]], _Finite sets and symmetric simplicial sets_ [PDF](http://www.tac.mta.ca/tac/volumes/8/n8/n8.pdf)

* [[Marco Grandis]], _Higher fundamental functors for simplicial sets_ [arXiv](http://arxiv.org/abs/math/0009004)

* [[Jiří Rosický]], [[Walter Tholen]], *Left-determined model categories and universal homotopy theories*, Trans. Amer. Math. Soc. 355 (2003), 3611-3623.

* [[Jiří Rosický]], [[Walter Tholen]], 2008, 'Erratum to "Left-determined model categories and universal homotopy theories"', _Transactions of the American Mathematical Society_, vol. 360, no. 11, pp. 6179-6179: doi:[10.1090/s0002-9947-08-04727-2](http://dx.doi.org/10.1090/s0002-9947-08-04727-2)

* [[Denis-Charles Cisinski]], _Les préfaisceaux comme modèles des types d'homotopie_, Astérisque 308, 2006 ([doi:10.24033/ast.715](https://smf.emath.fr/publications/les-prefaisceaux-comme-modeles-des-types-dhomotopie), [NUMDAM](http://www.numdam.org/item/AST_2006__308__R1_0/))

* Eric Ramón Antokoletz, _An Alternative Presentation of the Symmetric-Simplicial Category_, arXiv:1012.1587 (2010). ([abstract](https://arxiv.org/abs/1012.1587))

[[!redirects symmetric simplicial set]]
[[!redirects symmetric simplicial sets]]
[[!redirects symmetric sets]]