The rank invariant is of great interest in studying both one-dimensional and multidimensional persistence modules. Based on a category theoretical point of view, we generalize the rank invariant to zigzag modules. We prove that the rank invariant of a zigzag module recovers its interval decomposition. This proves that the rank invariant is a complete invariant for zigzag modules. The degree of difference between the rank invariants of any two zigzag modules can be measured via the erosion distance proposed by A.~Patel. We show that this erosion distance is bounded from above by the interleaving distance between the zigzag modules up to a multiplicative constant. Our construction allows us to extend the notion of   \rightarrow C from a category theory perspective. This generalizes the standard notion of rank invariant as well as Patel's recent extension. Specifically, the barcode of any interval decomposable persistence modules F:P \rightarrow vec of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion. Generalizing this idea allows freedom of choosing the indexing poset P of F: P \rightarrow C in defining Patel's } generalized persistence diagram by A. ~Patel to zigzag modules valued in a symmetric monoidal bicomplete category .
When a category \mathcal{C notion of rank invariant for arbitrary poset-indexed functors F:\mathbf{P \rightarrow C from a category theory perspective. This generalizes the standard notion of rank invariant as well as Patel's recent extension. Specifically, the barcode of any interval decomposable persistence modules F:P \rightarrow vec of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion. Generalizing this idea allows freedom of choosing the indexing poset P of F: P \rightarrow C in defining Patel's } generalized persistence diagram of F. By specializing our idea to zigzag persistence modules, we also show that the barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces. This leads to a promotion of Patel's semicontinuity theorem about type \mathcal{A .
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The rank invariant is of great interest in studying both one-dimensional and multidimensional persistence modules. Based on a category theoretical point of view, we generalize the rank invariant to zigzag modules. We prove that the rank invariant of a zigzag module recovers its interval decomposition. This proves that the rank invariant is a complete invariant for zigzag modules. The degree of difference between the rank invariants of any two zigzag modules can be measured via the erosion distance proposed by A.~Patel. We show that this erosion distance is bounded from above by the interleaving distance between the zigzag modules up to a multiplicative constant. Our construction allows us to extend the
When a category \mathcal{C
A
rank invariant
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for arbitrary poset-indexed functors F:\mathbf{P
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by A. ~Patel to zigzag modules valued in a symmetric monoidal bicomplete category
of F. By specializing our idea to zigzag persistence modules, we also show that the barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces. This leads to a promotion of Patel's semicontinuity theorem about type \mathcal{A