N-spherical functors and categorification of Euler's continuants
| Autor: | Dyckerhoff, Tobias, Kapranov, Mikhail, Schechtman, Vadim |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 infinity-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 in which case those of order N-2 are equivalences. The usual concept of a spherical functor corresponds to N=4. We characterize N-periodic semi-orthogonal decompositions of triangulated (stable infinity-) 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. Comment: 53 pages, 3 figures |
| Databáze: | arXiv |
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