#homotopy/stable-homotopy > Reference: Barnes and Roitzheim 7.1
A symmetric space \(F\) is a \({\mathsf{Top}}_*\) enriched functor \(F: \Sigma \to {\mathsf{Top}}_*\), where \(\Sigma\) is the category whose objects are \({\mathbb{N}}\) and \begin{align*}\Sigma(a, b)=\left\{\begin{array}{ll}\left(\Sigma_{a}\right)_{+} & \text {if } a=b \\ * & \text { if } a \neq b .\end{array}\right.\end{align*}
Equivalently, a sequence of pointed spaces \(X_n\) with associative unital maps \begin{align*} (\Sigma_n)_+ \wedge X_n \to X_{n} ,\end{align*} i.e. an action of \(\Sigma_n \curvearrowright X_n\).