https://bfluhr.com/at-tda-22/#title
| Object of Category | Functorial Invariant | Additive Invariant |
|---|---|---|
| Closed Manifold $X$ | Singular Chain Complex ${\Delta_{\bullet} (X)}$ | Euler Characteristic ${\chi(X) \in \Z}$ |
| Graded Singular Homology ${H_{\bullet} (X)}$ | Betti Numbers ${\beta_{\bullet} (X) \in \Z^{\N}}$ | |
| $\F$-tame Function ${f \colon X \to \R}$ | Relative Interlevel Set Cohomology (RISC) ${h(f) \colon \M^{\circ} \to \VectF}$ | Extended Persistence Diagram $\op{Dgm}(f)$ |
This way we obtain a functor ${h(f) \colon \M^{\circ} \rightarrow \VectF}$ vanishing on $\partial \M$, the relative interlevel set cohomology (RISC) of $f \colon X \rightarrow \R$.
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 $\mathcal{J}$.
The indecomposable ${B_v \colon \M^{\circ} \rightarrow \VectF}$
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 $\mathrm{pres}(\mathcal{J})$.