Hall (∞, 2)-Categories

Abstract

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 8-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 8-) 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.

On Frobenius exact symmetric tensor categories

“…Part (1) of the following proposition is essentially a special case of [DKS,Proposition 3.3.7]. We repeat the proof loc.…”

“…Remark 2.1.4. In general, it follows by construction, or from the detailed analysis in [DKS,§3], that E n (X) is a complex of the form…”

“…Proof. The definition of sphericality in [DKS,Definition 4.1.1] is given in terms of vanishing of certain (exact) functors on stable ∞-categories. These vanish if and only if they vanish on the homotopy category.…”

“…(2) In [DKS,§5.5], the notion of an 'N -spherical object' is introduced, generalising the notion of (4-)spherical objects in triangulated categories due to Seidel and Thomas. We therefore do not use the terminology (homotopically) 'N -spherical objects' for what we call (homotopically) 'N -bounded objects'.…”

“…The following result is a special case of [DKS,Proposition 4.1.4(b)], obtained via a completely different proof. We will henceforth say that X is strictly homotopically Nbounded if it is homotopically N -bounded, but not homotopically N ′ -bounded for N ′ < N .…”