Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the definition of homological critical value   We point out that there is a problem with the definition of homological
critical value (as defined in the widely cited paper \cite{stability} by
Cohen-Steiner, Edelsbrunner and Harer). Under that definition, the critical
value lemma of \cite{stability} in fact fails. We provide several
counterexamples and a definition (due to Bubenik and Scott
\cite{categorification}) we feel should be preferred and under which the
critical value lemma does indeed hold. One of the counterexamples we have found
is a height function on a compact smooth manifold. In the end we prove that,
despite all this, a modified version of the critical value lemma remains valid
under the original definition.

17
The categories of this paper are: q-fin
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con