Resonance category (Q1772476)

Given a pointed space \(X\), write \(X^{(n)}=\underbrace{X\wedge\cdots\wedge X}_n/S_n\) for the \(n\)-fold symmetric smash product of \(X\) with the canonical stratification, where \(S_n\) is the symmetric group. The author defines a new category \(\mathcal{R}\), called a resonance category to view that stratification of \(X^{(n)}\) as a a certain functor, called a resonance functor from \(\mathcal{R}\) to the category Top\(^\ast\) of pointed spaces. To illustrate this abstract framework, the author chooses the spaces of real (resp., complex) polynomials to study the Arnold problem [\textit{V. I.\ Arnold}, Trans.\ Moscow Math. Soc. 21(1970), 30--52 (1971), translation from Tr. Mosk. Mat. O.-va 21, 27--46 (1970; Zbl 0208.24003 )] of computing the algebraic invariants of these strata, for \(X=S^1\) (resp.\ \(X=S^2\)).