Categorification of Extended Persistence Diagrams

June 21, 2022

▴(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 \[ \dots \rightarrow F(T(u)) \rightarrow F(w) \rightarrow F(v) \rightarrow F(u) \rightarrow F(T^{-1}(w)) \rightarrow \dots \]


We say that a continuous function ${f \colon X \rightarrow \R}$ is $\F$-tame if ${\dim_{\F} H^n (f^{-1}(I); \F) < \infty}$ for all open intervals ${I \subseteq \R}$ and all ${n \in \Z.}$

For an $\F$-tame function $f \colon X \rightarrow \R$ its RISC ${h(f) \colon \M^{\circ} \rightarrow \VectF}$ has the following properties:

We denote the full subcategory of functors $\M^{\circ} \rightarrow \VectF$ vanishing on $\partial \M$ with these four properties by $\mathcal{J}$.

{0} {0} 𝔽 v ᴛ⁻¹(v)

The indecomposable ${B_v \colon \M^{\circ} \rightarrow \VectF}$


(Bauer, Botnan, F.)

Any functor ${F \colon \M^{\circ} \rightarrow \VectF}$ in $\mathcal{J}$ decomposes as \[ F \cong \bigoplus_{v \in \op{int} \M} (B_v)^{\oplus \nu(v)} \] for some ${\nu \colon \op{int} \M \rightarrow \N_0}$.


Any functor in $\mathcal{J}$ is projective in the full subcategory of functors ${\M^{\circ} \rightarrow \VectF}$ vanishing on $\partial \M$.

If ${F = h(f) \colon \M^{\circ} \rightarrow \VectF}$, then ${\nu \colon \op{int} \M \rightarrow \N_0}$ is the extended persistence diagram of ${f \colon X \rightarrow \R}$ due to Cohen-Steiner, Edelsbrunner and Harer.

The subcategory $\mathcal{J}$ of $\VectF^{\M^{\circ}}$ is not abelian.

Which full abelian subcategory of $\VectF^{\M^{\circ}}$ containing $\mathcal{J}$ is smallest?

Such a subcategory has to contain all cokernels in particular.


For an abelian category $\mathcal{C}$ and a full replete additive subcategory $\mathcal{J}$ of projectives in $\mathcal{C}$ we say that an object $X$ of $\mathcal{C}$ is $\mathcal{J}$-presentable if there is an exact sequence (called presentation) \[ Q \rightarrow P \rightarrow X \rightarrow 0 \] with $P$ and $Q$ in $\mathcal{J}$. We write $\mathrm{pres}(\mathcal{J})$ for the full subcategory of $\mathcal{J}$-presentable objects in $\mathcal{C}$.


The category of $\mathcal{J}$-presentable functors $\mathrm{pres}(\mathcal{J})$ is an abelian subcategory of $\VectF^{\M^{\circ}}$, which is also Frobenius with $\mathcal{J}$ being the subcategory of projective-injectives.

For any point $v \in \M$, there is an associated simple functor \[ S_v \colon \M^{\circ} \rightarrow \mathrm{Vect}_{\F},\, u \mapsto \begin{cases} \F & u = v \\ \{0\} & \text{otherwise} \end{cases} \] with all internal maps necessarily trivial.

Let $\mathcal{C}$ be the category of functors ${\M^{\circ} \rightarrow \VectF}$ vanishing on $\partial \M$.

By the previous theorem and corollary any functor ${F \colon \M^{\circ} \rightarrow \VectF}$ in $\mathrm{pres}(\mathcal{J})$ has a projective resolution in $\mathcal{C}$ by functors in $\mathcal{J}$.

Thus, we obtain finite-dimensional vector spaces ${\Ext_{\mathcal{C}}^n (F, S_v)}$.


For $\mathcal{J}$-presentable ${F \colon \M^{\circ} \rightarrow \VectF}$ its $n$-th Betti function is \[ \beta^n (F) \colon \op{int} \M \rightarrow \N_0, \, u \mapsto \dim_{\F} \mathrm{Ext}_{\mathcal{C}}^n (F, S_u) . \]

These Betti functions are closely related to the bigraded Betti numbers by Lesnick and Wright:


Any $\mathcal{J}$-presentable ${F \colon \M^{\circ} \rightarrow \VectF}$ admits a (potentially infinite) minimal projective resolution \[ \dots \rightarrow P_n \rightarrow \dots \rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow F \rightarrow 0 \] by functors ${P_n \colon \M^{\circ} \rightarrow \VectF}$ in $\mathcal{J}$. The multiplicity of ${B_u \colon \M^{\circ} \rightarrow \VectF}$ in $P_n$ is equal to ${\beta^n(F)(u) = \dim \Ext_{\mathcal{C}}^n (F, S_u)}$.

All Betti functions satisfy a certain finiteness constraint; we call functions $\op{int} \M \rightarrow \N_0$ satisfying this constraint admissible Betti functions.

We denote the commutative monoid of all admissible Betti functions by $\mathbb{B}$.

Moreover, the Euler function \[ \chi(F) % := \sum_{n \in \N_0} (-1)^n \beta^n(F) \colon \op{int} \M \rightarrow \Z, \, u \mapsto \sum_{n \in \N_0} (-1)^n \dim \Ext_{\mathcal{C}}^n(F, S_u) \] is well-defined and the pointwise absolute value \[ |\chi(F)| \colon \op{int} \M \rightarrow \N_0,\, u \mapsto |\chi(F)(u)| \] is an admissible Betti function as well, i.e. ${\chi(F) \colon \op{int} \M \rightarrow \Z}$ is an admissible Euler function.

Let $G(\mathbb{B})$ be the group of all admissible Euler functions.

The Euler function $\chi \colon \mathrm{Ob}(\mathrm{pres}(\mathcal{J})) \rightarrow G(\mathbb{B})$ is an additive invariant, hence there is an abelian group homomorphism \[ [\chi] \colon K_0(\mathrm{pres}(\mathcal{J})) \rightarrow G(\mathbb{B}), \, [F] \mapsto \chi(F) . \]


The group homomorphism ${[\chi] \colon K_0(\mathrm{pres}(\mathcal{J})) \rightarrow G(\mathbb{B})}$ is an isomorphism.

By the previous proposition ${\chi(h(f)) = \beta^0(h(f)) \colon \op{int} \M \rightarrow \Z}$ counts for each point $u \in \op{int} \M$ the multiplicity of ${B_u \colon \M^{\circ} \rightarrow \VectF}$.

This makes ${\chi(h(f)) \colon \op{int} \M \rightarrow \Z}$ the extended persistence diagram of ${f \colon X \rightarrow \R}$.

Thus, we may think of $\mathrm{pres}(\mathcal{J})$ as the categorification of extended persistence diagrams.

Derived level set persistence by Curry, Kashiwara, and Schapira: \[ R (-)_* \F_{(-)} \colon (\mathrm{Top} / \R)^{\circ} \rightarrow D^+(\R),\, (f \colon X \rightarrow \R) \mapsto R f_* \F_X . \]

In close analogy to the construction of RISC \[ h \colon (\mathrm{Top} / \R)^{\circ} \rightarrow \VectF^{\M^{\circ}} \] we may construct a functor \[ h_{\R} \colon D^+(\R) \rightarrow \VectF^{\M^{\circ}} \] in terms of local sheaf cohomology in place of singular cohomology.

Using the isomorphism between sheaf cohomology and singular cohomology for locally contractible spaces one can describe a natural isomorphism $\zeta$:


We say that an object $F$ of $D^+(\R)$ is tame if $H^n(I; F)$ is finite-dimensional for any open interval $I \subseteq \R$; denoting the full subcategory of tame objects by $D^+_t (\R)$.


The functor $h_{\R}$ restricts to an equivalence of categories ${h_{\R,t} \colon D^+_t (\R) \rightarrow \mathcal{J}}$.

Thus, we obtain the diagram with the lower horizontal functor an equivalence of categories.

In conjunction with a slight generalization of a result by Krause we obtain:


The composition of functors \[ D^+_t (\R) \xrightarrow{h_{\R,t}} \mathcal{J} \hookrightarrow \mathrm{pres}(\mathcal{J}) \] is the abelianization of $D^+_t (\R)$.

By the first proposition any $\mathcal{J}$-presentable admits a projective cover by a functor in $\mathcal{J}$.

Thus, indeed $\mathcal{J}$ is the subcategory of projectives in $\mathrm{pres}(\mathcal{J})$.