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- W3099327281 abstract "For an additive category <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=bold>P</mml:mi></mml:mrow></mml:math> we provide an explicit construction of a category <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>Q</mml:mi></mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=bold>P</mml:mi></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:math> whose objects can be thought of as formally representing <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mfrac><mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mtext>im</mml:mtext></mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mi>γ</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mtext>im</mml:mtext></mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mi>ρ</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>∩</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mtext>im</mml:mtext></mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mi>γ</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:mfrac></mml:math> for given morphisms <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>γ</mml:mi><mml:mo>:</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy=false>→</mml:mo><mml:mi>B</mml:mi></mml:math> and <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>ρ</mml:mi><mml:mo>:</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=false>→</mml:mo><mml:mi>B</mml:mi></mml:math> in <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=bold>P</mml:mi></mml:mrow></mml:math>, even though <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=bold>P</mml:mi></mml:mrow></mml:math> does not need to admit quotients or images. We show how it is possible to calculate effectively within <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>Q</mml:mi></mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=bold>P</mml:mi></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:math>, provided that a basic problem related to syzygies can be handled algorithmically. We prove an equivalence of <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>Q</mml:mi></mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=bold>P</mml:mi></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:math> with the smallest subcategory of the category of contravariant functors from <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=bold>P</mml:mi></mml:mrow></mml:math> to the category of abelian groups <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=bold>A</mml:mi><mml:mi mathvariant=bold>b</mml:mi></mml:mrow></mml:math> which contains all finitely presented functors and is closed under the operation of taking images. Moreover, we characterize the abelian case: <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>Q</mml:mi></mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=bold>P</mml:mi></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:math> is abelian if and only if it is equivalent to <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=normal>f</mml:mi><mml:mi mathvariant=normal>p</mml:mi></mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:msup><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=bold>P</mml:mi></mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=normal>o</mml:mi><mml:mi mathvariant=normal>p</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=bold>A</mml:mi><mml:mi mathvariant=bold>b</mml:mi></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:math>, the category of all finitely presented functors, which in turn, by a theorem of Freyd, is abelian if and only if <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=bold>P</mml:mi></mml:mrow></mml:math> has weak kernels.The category <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>Q</mml:mi></mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=bold>P</mml:mi></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:math> is a categorical abstraction of the data structure for finitely presented <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>R</mml:mi></mml:math>-modules employed by the computer algebra system Macaulay2, where <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>R</mml:mi></mml:math> is a ring. By our generalization to arbitrary additive categories, we show how this data structure can also be used for modeling finitely presented graded modules, finitely presented functors, and some not necessarily finitely presented modules over a non-coherent ring." @default.
- W3099327281 created "2020-11-23" @default.
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- W3099327281 date "2020-09-01" @default.
- W3099327281 modified "2023-09-27" @default.
- W3099327281 title "Closing the category of finitely presented functors under images made constructive" @default.
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- W3099327281 doi "https://doi.org/10.32408/compositionality-2-4" @default.
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