Generalized persistence diagrams

Abstract

We generalize the persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer to the setting of constructible persistence modules valued in a symmetric monoidal category. We call this the type ${\mathcal {A}}$ persistence diagram of a persistence module. If the category is also abelian, then we define a second type ${\mathcal {B}}$ persistence diagram. In addition, we show that both diagrams are stable to all sufficiently small perturbations of the module. The type ${\mathcal {B}}$ persistence diagram carries less information than the type ${\mathcal {A}}$ persistence diagram, but it enjoys a stronger stability theorem.

Appendix: Krull-Schmidt

We now provide a compact treatment of Krull-Schmidt categories. The following ideas are classical and may be found in many books, for example Anderson and Fuller (1992).

A category ${\mathsf {C}}$ is additive if all its hom-sets are abelian, composition is bilinear, and finite products and finite coproducts are the same. The (co)product of the empty set is the zero object of ${\mathsf {C}}$. Suppose ${\mathsf {C}}$ is additive.

Definition A.1

A non-zero object $a \in {\mathsf {C}}$ is indecomposable if it is not the direct sum of two non-zero objects.

Definition A.2

An additive category ${\mathsf {C}}$ is Krull-Schmidt if each object $a \in {\mathsf {C}}$ is isomorphic to a finite direct sum $a \cong a_1 \oplus a_2 \oplus \cdots \oplus a_n$ and each ring of endomorphisms ${\mathsf {End}}_{\mathsf {C}}(a_i)$ is local. That is, $0 \ne 1$ and if $f_1 + f_2 = 1$, then $f_1$ or $f_2$ is invertible.

Suppose ${\mathsf {C}}$ is Krull-Schmidt.

Proposition 9.1

An object $a \in {\mathsf {C}}$ is indecomposable iff its endomorphism ring ${\mathsf {End}}(a)$ is local.

Proof

Suppose $a \in {\mathsf {C}}$ is decomposable. That is, there is an isomorphism $i :a \rightarrow a_1 \oplus a_2$ such that $a_1,a_2 \ne 0$. Define $\pi _1 : a_1 \oplus a_2 \rightarrow a_1 \oplus a_2$ as the endomorphism that sends the first factor to zero and $\pi _2 : a_1 \oplus a_2 \rightarrow a_1 \oplus a_2$ as the endomorphism that sends the second factor to zero. Then the two maps $\rho _1, \rho _2 : a \rightarrow a$, where $\rho _1 = i^{-1} \circ \pi _1 \circ i$ and $\rho _2 = i^{-1} \circ \pi _2 \circ i$, are both non-isomorphisms in ${\mathsf {End}}_{\mathsf {C}}(a)$. However, $\rho _0 + \rho _1 :a \rightarrow a$ is an isomorphism. We have a contradiction of locality.

Suppose $a \in {\mathsf {C}}$ is indecomposable. Then, by definition of a Krull-Schmidt category, ${\mathsf {End}}_{\mathsf {C}}(a)$ is a local ring. $\square$

Proposition 9.2

Each object $a \in {\mathsf {C}}$ is isomorphic to a finite direct sum of indecomposables.

Proof

By definition of a Krull-Schmidt category, $a \cong a_1 \oplus a_2 \oplus \cdots \oplus a_n$ where each ${\mathsf {End}}_{\mathsf {C}}(a_i)$ is a local ring. By Proposition 9.1, each $a_i$ is indecomposable. $\square$

Theorem 9.1

(Krull-Schmidt) Suppose an object $c \in {\mathsf {C}}$ is isomorphic to $a_1 \oplus a_2 \oplus \cdots \oplus a_m$ and $b_1 \oplus b_2 \oplus \cdots \oplus b_n$, where each $a_i$ and $b_j$ are indecomposable. Then $m = n$, and there is a permutation $p: [m] \rightarrow [n]$ such that $a_i \cong b_{p(i)}$.

Proof

By definition of an additive category, we have canonical projections $\pi _i : \oplus _i a_i \rightarrow a_i$ and $\rho _j : \oplus _j b_j \rightarrow b_j$ and canonical inclusions $\mu _i : a_i \rightarrow \oplus _i a_i$ and $\nu _j : b_j \rightarrow \oplus _j b_j$. Furthermore $\mu _j \circ \pi _i$ and $\nu _j \circ \rho _i$ are the identity on $a_i$ and $b_i$, respectively, iff $i = j$. Let $f : a_1 \oplus a_2 \oplus \cdots \oplus a_m \rightarrow b_1 \oplus b_2 \oplus \cdots \oplus b_n$ be an isomorphism.

Define $h_j : a_1 \rightarrow a_1$ as $h_j = \pi _1 \circ f^{-1} \circ \nu _j \circ \rho _j \circ f \circ \mu _1$. Let $h = \sum _j h_j: a_1 \rightarrow a_1$. Observe h is an isomorphism. By locality, there is an index j such that $h_j$ is an isomorphism. This means $a_1 \cong b_j$ and we specify $p(1) = j$. Quotient by $a_1$ and $b_j$. Repeat. $\square$