Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Overlays and group actions   Overlays were introduced by R. H. Fox [6] as a subclass of covering maps. We
offer a different view of overlays: it resembles the definition of paracompact
spaces via star refinements of open covers. One introduces covering structures
for covering maps and $p:X\to Y$ is an overlay if it has a covering structure
that has a star refinement.
  We prove two characterizations of overlays: one using existence and
uniqueness of lifts of discrete chains, the second as maps inducing simplicial
coverings of nerves of certain covers. We use those characterizations to
improve results of Eda-Matijevi\' c concerning topological group structures on
domains of overlays whose range is a compact topological group.
  In case of surjective maps $p:X\to Y$ between connected metrizable spaces we
characterize overlays as local isometries: $p:X\to Y$ is an overlay if and only
if one can metrize $X$ and $Y$ in such a way that $p|B(x,1):B(x,1)\to
B(p(x),1)$ is an isometry for each $x\in X$.

5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin