Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Auslander--Reiten theory in extriangulated categories   The notion of an extriangulated category gives a unification of existing
theories in exact or abelian categories and in triangulated categories. In this
article, we develop Auslander--Reiten theory for extriangulated categories.
This unifies Auslander--Reiten theories developed in exact categories and
triangulated categories independently. We give two different sets of sufficient
conditions on the extriangulated category so that existence of almost-split
extensions becomes equivalent to that of an Auslander--Reiten--Serre duality.
We also show that existence of almost-split extensions is preserved under
taking relative extriangulated categories, ideal quotients, and
extension-closed subcategories. Moreover, we prove that the stable category
$\underline{\mathscr{C}}$ of an extriangulated category $\mathscr{C}$ is a
$\tau$-category if $\mathscr{C}$ has almost split extensions and source
morphisms. This gives various consequences on $\underline{\mathscr{C}}$,
including Igusa--Todorov's Radical Layers Theorem. In particular, the
associated completely graded category of $\underline{\mathscr{C}}$ is
equivalent to the complete mesh category of the Auslander-Reiten species of
$\underline{\mathscr{C}}$.

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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: ao-sci