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Theory of interleavings on categories with a flow

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V. de Silva, E. Munch, A. Stefanou

The interleaving distance was originally defined in the field of
Topological Data Analysis (TDA) by Chazal et al. as a metric on the
class of persistence modules parametrized over the real line. Bubenik et
al. subsequently extended the definition to categories of functors on a
poset, the objects in these categories being regarded as `generalized
persistence modules'. These metrics typically depend on the choice of a
lax semigroup of endomorphisms of the poset. The purpose of the present
paper is to develop a more general framework for the notion of
interleaving distance using the theory of `actegories'. Specifically, we
extend the notion of interleaving distance to arbitrary categories
equipped with a flow, i.e. a lax monoidal action by the monoid
$[0,\infty)$. In this way, the class of objects in such a category
acquires the structure of a Lawvere metric space. Functors that are
colax $[0,\infty)$-equivariant yield maps that are 1-Lipschitz. This
leads to concise proofs of various known stability results from TDA, by
considering appropriate colax $[0,\infty)$-equivariant functors. Along
the way, we show that several common metrics, including the Hausdorff
distance and the $L^\infty$-norm, can be realized as interleaving
distances in this general perspective.

Keywords:
Topological Data Analysis, Persistent Homology, Category Theory, Lawvere
Metric Spaces

2010 MSC:
18C10, 18D05, 18D10, 18D20 and 55N10

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 21, pp 583-607.

Published 2018-06-05.

http://www.tac.mta.ca/tac/volumes/33/21/33-21.pdf

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