Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A motivic Segal theorem for open pairs of smooth schemes over an
  infinite perfect field   V. Voevodsyky laid the groundwork of delooping motivic spaces in order to
provide a new, more computation-friendly, construction of the stable motivic
category $SH(k)$, G. Garkusha and I. Panin made that project a reality, while
collaborating with A. Ananievsky, A. Neshitov and A. Druzhinin. In particular,
G. Garkusha and I. Panin proved that for an infinite perfect field $k$ and any
$k$-smooth scheme $X$ the canonical morphism of motivic spaces $C_*Fr(X)\to
\Omega^{\infty}_{\mathbb{P}^1} \Sigma^{\infty}_{\mathbb{P}^1} (X_+)$ is
Nisnevich-locally a group-completion.
  In the present work, a generalisation of that theorem to the case of smooth
open pairs $(X,U),$ where $X$ is a $k$-smooth scheme, $U$ is its open subscheme
intersecting each component of $X$ in a nonempty subscheme. We claim that in
this case the motivic space $C_*Fr((X,U))$ is Nisnevich-locally connected, and
the motivic space morphism $C_*Fr((X,U))\to \Omega^{\infty}_{\mathbb{P}^1}
\Sigma^{\infty}_{\mathbb{P}^1} (X/U)$ is Nisnevich-locally a weak equivalence.
Moreover, we show that if the codimension of $S=X-U$ in each component of $X$
is greater than $r \geq 0,$ the simplicial sheaf $C_*Fr((X,U))$ is locally
$r$-connected.

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The categories of this paper are: q-bio
36
The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: math-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ph
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The categories of this paper are: stat
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con