Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Pullback diagrams, syzygy finite classes and Igusa-Todorov algebras   For an abelian category $\mathcal{A}$, we define the category
PEx($\mathcal{A}$) of pullback diagrams of short exact sequences in
$\mathcal{A}$, as a subcategory of the functor category Fun($\Delta,
\mathcal{A}$) for a fixed diagram category $\Delta$. For any object $M$ in
${\rm PEx}(\mathcal{A}),$ we prove the existence of a short exact sequence $0
{\to} K {\to} P {\to} M {\to} 0$ of functors, where the objects are in
PEx($\mathcal{A}$) and $P(i) \in {\rm Proj(\mathcal{A})}$ for any $i \in
\Delta$. As an application, we prove that if $(\mathcal{C}, \mathcal{D},
\mathcal{E})$ is a triple of syzygy finite classes of objects in
$\mathrm{mod}\,\Lambda$ satisfying some special conditions, then $\Lambda$ is
an Igusa-Todorov algebra. Finally, we study lower triangular matrix Artin
algebras and determine in terms of their components, under reasonable
hypothesis, when these algebras are syzygy finite or Igusa-Todorov.

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The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
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The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
1
The categories of this paper are: cs
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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: supr-con