Categorification of (Extended) Persistence Diagrams

December 13th, 2022

0 1 a c b A B

\[ \begin{split} f(0) = f(1) &= 0 = f(a) = f(b) \\ f(c) &= 1 = f(B) \\ f(A) &= 2 \end{split} \]

We obtain a pointwise Mayer–Vietoris sequence in persistent homology

and from this a short exact sequece

\[ 0 \rightarrow \op{coker} \varphi_n \rightarrow h_n (f) \rightarrow \ker \varphi_{n-1} \rightarrow 0 \]

of persistence modules for each $n \in \N_0$.

▴(a) ▴(b) H⁰(f⁻¹(ℝ, ℝ∖[a, b])) H⁰(f⁻¹(ℝ, (-∞,a))) 𝕄 H⁰(f⁻¹((-∞,b), (-∞,a))) H⁰(f⁻¹(ℝ, (b,∞))) H⁰(f⁻¹(-∞,b)) H⁰(f⁻¹((a,∞), (b,∞))) H⁰(f⁻¹(a,b)) ʜ¹ ʜ² ʜ³ -∞
T(u) u

This way we obtain a functor ${h(f) \colon \M^{\circ} \rightarrow \VectF}$ vanishing on $\partial \M$, the relative interlevel set cohomology (RISC) of $f \colon X \rightarrow \R$.

ᴛ³(u) ᴛ²(w) ᴛ²(v) ᴛ²(u) ᴛ(w) ᴛ(v) ᴛ(u) w v u $l_1$ $l_0$

Cohomological functor $F \colon \M^{\circ} \rightarrow \VectF$ yields long exact sequence \[ \dots \rightarrow F(T(u)) \rightarrow F(w) \rightarrow F(v) \rightarrow F(u) \rightarrow F(T^{-1}(w)) \rightarrow \dots \]

a b c
▴ c ▴ b ▴ a
Rel₂ Ord₀ Rel₀ Rel₁ Ord₁ Ext₂ Ext₁ Ext₀

As a counterpart to the Mayer–Vietoris sequence we obtain the exact sequence in RSSC

\[ \begin{CD} h'(f |_A) \circ T \oplus h'(f |_B) \circ T \\ @VV{\varphi \circ T}V \\ h'(f |_{A \cap B}) \circ T \\ @VVV \\ h'(f) \\ @VVV \\ h'(f |_A) \oplus h'(f |_B) \\ @VV{\varphi}V \\ h'(f |_{A \cap B}) \end{CD} \]

as well as the induced short exact sequence of presheaves on $\M'$:

\[ 0 \rightarrow \op{coker} \varphi \circ T \rightarrow h'(f) \rightarrow \ker \varphi \rightarrow 0 . \]

How can we assign a persistence diagram to the presheaves $\op{coker} \varphi \circ T$ and $\ker \varphi$ in a meaningful way?

Observation: Defining the simple presheaf \[ S_v \colon \M'^{\circ} \rightarrow \mathrm{Vect}_{\F},\, u \mapsto \begin{cases} \F & u = v \\ \{0\} & \text{otherwise} \end{cases} \] we have $\op{Dgm}'(f)(v) = \dim \mathrm{Nat}(h'(f), S_v).$

Now $\dim$ is additive, but $\mathrm{Nat}(-, S_v)$ is only left-exact.

So we define the Betti function \[\beta^n (F) \colon \op{int} \M' \rightarrow \N_0, \, u \mapsto \dim_{\F} \mathrm{Ext}_{\mathcal{C}}^n (F, S_u)\]

as well as the Euler function $\chi(F) := \sum_{n \in \N_0} (-1)^n \beta^n(F)$ where ...

... $\mathcal{C}$ is the category of presheaves on $\M'$ vanishing on $\partial \M'$.

Thank You!