Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Stabilization and costabilization with respect to an action of a
  monoidal category   We study actions of monoidal categories on objects in a suitably enriched
$2$-category, and applications in stable homotopy theory. Given a monoidal
category $\mathcal{I}$ and an $\mathcal{I}$-object $\mathcal{A}$, the
(co)stabilization of $\mathcal{A}$ is obtained by universally forcing the
$\mathcal{I}$-action to be reversible so that every object of $\mathcal{I}$
acts on $\mathcal{A}$ by auto-equivalences. We introduce a notion of
$\mathcal{I}$-equivariance for morphisms between $\mathcal{I}$-objects and give
constructions of stabilization and costabilization in terms of weak ends and
coends in an enriched $2$-category of $\mathcal{I}$-objects and
$\mathcal{I}$-equivariant morphisms. We observe that the stabilization of a
relative category with respect to an action coincides with the usual notion of
stabilization in stable homotopy theory when the action is defined by loop
space functors. We show that several examples that exist in the literature,
including various categories of spectra, fit into our setting after fixing
$\mathcal{A}$ and the $\mathcal{I}$-action on it. In particular, categories of
sequential spectra, coordinate free spectra, genuine equivariant spectra,
parameterized spectra indexed by vector bundles are obtained in terms of weak
ends in the $2$-category of relative categories. On the other hand, the
costabilization of a relative category with respect to an action gives a stable
relative category akin to a version Spanier-Whitehead category. In particular,
we establish a form of duality between constructions of stable homotopy
categories by spectra and by Spanier-Whitehead like categories.

27
The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
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The categories of this paper are: q-alg