Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An Approximate Nerve Theorem   The Nerve Theorem relates the topological type of a suitably nice space with
the nerve of a good cover of that space. It has many variants, such as to
consider acyclic covers and numerous applications in topology including applied
and computational topology. The goal of this paper is to relax the notion of a
good cover to an approximately good cover, or more precisely, we introduce the
notion of an $\varepsilon$-acyclic cover. We use persistent homology to make
this rigorous and prove tight bounds between the persistent homology of a space
endowed with a function and the persistent homology of the nerve of an
$\varepsilon$-acyclic cover of the space. Using the Mayer-Vietoris spectral
sequence, we upper bound how local non-acyclicity can affect the global
homology. To prove the best possible bound we must introduce special cases of
interleavings between persistence modules called left and right interleavings.
Finally, we provide examples which achieve the bound proving the lower bound
and tightness of the result.

25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
24
The categories of this paper are: dg-ga
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org