Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A hierarchy of maps between compacta   Let $CH$ be the class of compacta (i.e., compact Hausdorff spaces), with $BS$
the subclass of Boolean spaces. For each ordinal $alpha$ and pair $(K,L)$ of
subclasses of $CH$, we define $Lev_{>=alpha}(K,L)$, the class of maps of level
at least $alpha$ from spaces in $K$ to spaces in $L$, in such a way that, when
$alpha < omega$, $Lev}_{>=alpha}(BS,BS)$ consists of the Stone duals of Boolean
lattice embeddings that preserve all prenex first-order formulas of quantifier
rank $alpha$. Maps of level $>=0$ are just the continuous surjections, and the
maps of level $>=1$ are the co-existential maps. Co-elementary maps are of
level $>=alpha$ for all ordinals $alpha$; of course in the Boolean context, the
co-elementary maps coincide with the maps of level $>=omega$. The results of
this paper include:
  (i) every map of level $>=omega$ is co-elementary;
  (ii) the limit maps of an $omega$-indexed inverse system of maps of level
$>=alpha$ are also of level $>=alpha$; and
  (iii) if $K$ is a co-elementary class, $k < omega$ and $Lev_{>=k}(K,K) =
Lev_{>=k+1}(K,K)$, then $Lev_{>=k}(K,K}) = Lev_{>=omega}(K,K)$.
  A space $X$ in $K$ is co-existentially closed in $K$ if $Lev_{>=0}(K,{X}) =
Lev_{>=1}(K,{X})$. We showed in an earlier paper that every infinite member of
a co-inductive co-elementary class (such as $CH$ itself, $BS$, or the class
$CON$ of continua) is a continuous image of a space of the same weight that is
co-existentially closed in that class. We show here that every compactum that
is co-existentially closed in $CON$ (a "co-existentially closed continuum") is
both indecomposable and of covering dimension one.

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The categories of this paper are: acc-phys
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The categories of this paper are: eess
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The categories of this paper are: cs
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-th
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The categories of this paper are: astro-ph
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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ph
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The categories of this paper are: q-bio