Let \(\mathfrak{Top}\) denote the category of (compactly generated weak Hausdorff) topological spaces. Throughout the paper, by a (topological) space the authors refer to an object in \(\mathfrak{Top}\), and by a persistent space they mean a functor from the poset category \((\mathbb{R},\leq)\) to \(\mathfrak{Top}\). This paper considers only persistent spaces with a discrete set of critical values. In addition, all (co)homology groups are assumed to be taken over a field \(K\). They denote by \(\mathfrak{Int}_\omega\) the set of intervals of type \(\omega\), where \(\omega\) can be any one of the four types: open-open, open-closed, closed-open and closed-closed. Results in this paper apply to all four situations, so for simplicity of notation, they state their results only for closed-closed intervals and omit \(\omega\) unless otherwise stipulated.
Let \((\mathfrak{n},\leq)\) be a poset category with a partial order \(\leq\). For any given category \(\mathcal{C}\), they define the \(\mathfrak{n}\)-valued categorical invariants to be maps \(\text{I}:\text{Ob}(\mathcal{C})\sqcup \text{Mor}(\mathcal{C})\to\mathfrak{n}\) assigning values to both objects and morphisms in \(\mathcal{C}\), such that \(\text{I}(\text{id}_X)=\text{I}(X)\) for all \(X\in \text{Ob}(\mathcal{C})\) and \(\text{I}(g\circ f)\leq\min\{\text{I}(f),\text{I}(g)\}\) for any \(f:X\to Y, g: Y\to Z\) in \(\mathcal{C}\). A persistent object in \(\mathcal{C}\) is any functor \(F_\bullet:(\mathbb{R},\leq)\to\mathcal{C}\).
Here \(\mathfrak{Vec}\) denotes the category of finite-dimensional vector spaces over a given field \(K\).
For a persistent space \(X_\bullet:(\mathbb{R},\leq)\to\mathfrak{Top}\) with \(t\mapsto X_t\), the persistent cup-length invariant \(\text{cup}(X_\bullet):\mathfrak{Int}\to\mathbb{N}\) of \(X_\bullet\), is defined as the functor from \((\mathfrak{Int},\subseteq)\) to \((\mathbb{N},\geq)\) of non-negative integers, which assigns to each interval \([a,b]\) the cup-length of the image ring \(\text{im}\left(\text{H}^\ast(X_b)\to \text{H}^\ast(X_a)\right)\).
They define the persistent LS-category invariant of a persistent space \(X_\bullet\) to be the function \(\text{cat}(X_\bullet):\mathfrak{Int}\to\mathbb{N}\) of \(X_\bullet\) assigning to each interval \([a,b]\) the LS-category of the transition map \(X_a\to X_b\).
Let \(d_{\text{E}}\) be the erosion distance, \(d_{\text{I}}\) be the interleaving distance, \(d_{\text{HI}}\) be the homotopy-interleaving distance and \(d_{\text{GH}}\) be the Gromov-Hausdorff distance.
The main results in the present article are as follows.
Theorem 1 (\(d_{\text{I}}\)-stability of persistent invariants) Let \(\mathcal{C}\) be a category, and let \(\text{I}:\text{Ob}(\mathcal{C})\sqcup \text{Mor}(\mathcal{C})\to\mathfrak{n}\) be a categorical invariant of \(\mathcal{C}\). The persistent \(I\)-invariant is I-Lipschitz stable, that is, for any persistent objects \(F_\bullet,G_\bullet:(\mathbb{R},\leq)\to\mathcal{C}\), \[ d_{\text{E}}(I(F_\bullet),I(G_\bullet))\leq d_{\text{I}}(F_\bullet,G_\bullet). \] Theorem 2 (Homotopical stability) Let \(I\) be a categorical invariant of topological spaces satisfying the condition that for any maps \(X\stackrel{f}{\to} Y\stackrel{g}{\to} Z\stackrel{h}{\to} W\) where \(g\) is a weak homotopy equivalence, \(I(g\circ f)=I(f)\) and \(I(h\circ g)=I(h)\). Then, for two persistent spaces \(X_\bullet,Y_\bullet:(\mathbb{R},\leq)\to \mathfrak{Top}\), we have \[ d_{\text{E}}(I(X_\bullet),I(Y_\bullet))\leq d_{\text{HI}}(X_\bullet,Y_\bullet). \]
For the Vietoris-Rips filtrations \(VR_\bullet(X)\) and \(VR_\bullet(Y)\) of compact metric spaces \(X\) and \(Y\), we have \[ d_{\text{E}}(I(VR_\bullet(X)),I(VR_\bullet(Y)))\leq 2\cdot d_{\text{GH}}(X,Y). \] A standard persistence module \(M_\bullet:(\mathbb{R},\leq)\to \mathfrak{Vec}\) is called \(q\)-tame if it satisfies the condition that \(\text{rk}\left(M_t\to M_{t'}\right)<\infty\) whenever \(t<t'\).
Theorem 3 For persistent spaces \(X_\bullet\) and \(Y_\bullet\) with \(q\)-tame persistent (co)homology, we have \[ d_{\text{E}}(\text{cup}(X_\bullet),\text{cup}(Y_\bullet))\leq d_{\text{E}}(\text{rk}(\Phi(X_\bullet)),\text{rk}(\Phi(Y_\bullet)))\leq d_{\text{HI}}(X_\bullet,Y_\bullet). \] For the Vietoris-Rips filtrations \(VR_\bullet(X)\) and \(VR_\bullet(Y)\) of two metric spaces \(X\) and \(Y\) all the above quantities are bounded above by \(2\cdot d_{\text{GH}}(X,Y)\).