Euler’s continuants are universal polynomials expressing the numerator and denominator of a finite continued fraction whose entries are independent variables. We introduce their categorical lifts which are natural complexes (more precisely, coherently commutative cubes) of functors involving compositions of a given functor and its adjoints of various orders, with the differentials built out of units and counits of the adjunctions. In the stable <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-categorical context these complexes/cubes can be assigned totalizations which are new functors serving as higher analogs of the spherical twist and cotwist. We define N-spherical functors by vanishing of the twist and cotwist of order <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> in which case those of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$N-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> are equivalences. The usual concept of a spherical functor corresponds to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N=4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>. We characterize N-periodic semi-orthogonal decompositions of triangulated (stable <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-) categories in terms of N-sphericity of their gluing functors. The procedure of forming iterated orthogonals turns out to be analogous to the procedure of forming a continued fraction.

N-spherical functors and categorification of Euler’s continuants

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2026年2月12日

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Tobias Dyckerhoff,
Mikhail Kapranov,
Vadim Schechtman

摘要

Euler’s continuants are universal polynomials expressing the numerator and denominator of a finite continued fraction whose entries are independent variables. We introduce their categorical lifts which are natural complexes (more precisely, coherently commutative cubes) of functors involving compositions of a given functor and its adjoints of various orders, with the differentials built out of units and counits of the adjunctions. In the stable ${\infty }$ $\infty$ -categorical context these complexes/cubes can be assigned totalizations which are new functors serving as higher analogs of the spherical twist and cotwist. We define N-spherical functors by vanishing of the twist and cotwist of order $N-1$ $N - 1$ in which case those of order $N-2$ $N - 2$ are equivalences. The usual concept of a spherical functor corresponds to $N=4$ $N = 4$ . We characterize N-periodic semi-orthogonal decompositions of triangulated (stable ${\infty }$ $\infty$ -) categories in terms of N-sphericity of their gluing functors. The procedure of forming iterated orthogonals turns out to be analogous to the procedure of forming a continued fraction.