<p>In this paper we introduce and study <i>rectangular torsion theories</i>, i.e. those torsion theories <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(\mathbb {C},\mathcal {T},\mathcal {F})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo>,</mo> <mi mathvariant="script">T</mi> <mo>,</mo> <mi mathvariant="script">F</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation> a pointed category, where the canonical functor <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {C}\rightarrow \mathcal {T}\times \mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">T</mi> <mo>×</mo> <mi mathvariant="script">F</mi> </mrow> </math></EquationSource> </InlineEquation> is an equivalence of categories. We prove that rectangular torsion theories are pseudo-algebras for a suitable 2-monad on the 2-category of pointed categories. Interestingly, it turns out that this 2-monad is a 2-categorical counterpart of the monad for rectangular bands, i.e., non-empty semigroups satisfying <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$xyz = xz$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mi>z</mi> </mrow> </math></EquationSource> </InlineEquation>. So rectangular torsion theories can be viewed as internal rectangular bands in the 2-category of pointed categories. We also show that a triple <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(\mathbb {C},\mathcal {T},\mathcal {F})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo>,</mo> <mi mathvariant="script">T</mi> <mo>,</mo> <mi mathvariant="script">F</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a rectangular torsion theory if and only if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation> is a pointed category with full replete subcategories <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {T},\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">T</mi> <mo>,</mo> <mi mathvariant="script">F</mi> </mrow> </math></EquationSource> </InlineEquation> such that: (i) every morphism between an object in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">T</mi> </math></EquationSource> </InlineEquation> and an object in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> (in either direction) is a zero morphism; (ii) any object in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation> decomposes as a biproduct of an object in <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">T</mi> </math></EquationSource> </InlineEquation> and an object in <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>, and, all biproducts of an object in <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathcal {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">T</mi> </math></EquationSource> </InlineEquation> with an object in <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> exist.</p>

Rectangular Torsion Theories

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2026年7月4日

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Elena Caviglia,
Zurab Janelidze,
Luca Mesiti

摘要

In this paper we introduce and study rectangular torsion theories, i.e. those torsion theories $(\mathbb {C},\mathcal {T},\mathcal {F})$ $(C, T, F)$ with $\mathbb {C}$ $C$ a pointed category, where the canonical functor $\mathbb {C}\rightarrow \mathcal {T}\times \mathcal {F}$ $C \to T \times F$ is an equivalence of categories. We prove that rectangular torsion theories are pseudo-algebras for a suitable 2-monad on the 2-category of pointed categories. Interestingly, it turns out that this 2-monad is a 2-categorical counterpart of the monad for rectangular bands, i.e., non-empty semigroups satisfying $xyz = xz$ $x y z = x z$ . So rectangular torsion theories can be viewed as internal rectangular bands in the 2-category of pointed categories. We also show that a triple $(\mathbb {C},\mathcal {T},\mathcal {F})$ $(C, T, F)$ is a rectangular torsion theory if and only if $\mathbb {C}$ $C$ is a pointed category with full replete subcategories $\mathcal {T},\mathcal {F}$ $T, F$ such that: (i) every morphism between an object in $\mathcal {T}$ $T$ and an object in $\mathcal {F}$ $F$ (in either direction) is a zero morphism; (ii) any object in $\mathbb {C}$ $C$ decomposes as a biproduct of an object in $\mathcal {T}$ $T$ and an object in $\mathcal {F}$ $F$ , and, all biproducts of an object in $\mathcal {T}$ $T$ with an object in $\mathcal {F}$ $F$ exist.