Generalised Moore spectra in a triangulated category

A note on triangulated monads and categories of module spectra

Consider a monad on an idempotent complete triangulated category with the property that its Eilenberg-Moore category of modules inherits a triangulation. We show that any other triangulated adjunction realizing this monad is 'essentially monadic', i.e. becomes monadic after performing the two evident necessary operations of taking the Verdier quotient by the kernel of the right adjoint and idempotent completion. In this sense, the monad itself is 'intrinsically monadic'. It follows that for any highly structured ring spectrum, its category of homotopy (a.k.a. naive) modules is triangulated if …

广义辫群中的范畴化与动力学

Categorification and Dynamics in Generalised Braid Groups

该博士论文受三角范畴与曲面的关联启发，通过范畴化Burau表示将秩2广义辫群实现为三角范畴自等价群，借助范畴熵研究其动力学，用类似映射类群分类的方法完成分类并给出可计算质量增长的算法。

Nearly Morita equivalences and rigid objects

If two cluster-tilting objects of an acyclic cluster category are related by a mutation, then their endomorphism algebras are nearly-Morita equivalent [Buan-Marsh-Reiten], i.e. their module categories are equivalent "up to a simple module". This result has been generalised by D. Yang, using a result of P-G. Plamondon, to any simple mutation of maximal rigid objects in a 2-Calabi--Yau triangulated category. In this paper, we investigate the more general case of any mutation of a (non-necessarily maximal) rigid object in a triangulated category with a Serre functor. In that setup, the endomorphi…

Classification of co-slicings and co-t-structures for the Kronecker algebra

In this paper we introduce the notion of a 'generalised' co-slicing of a triangulated category. This generalises the theory of co-stability conditions in a manner analogous to the way in which Gorodentsev, Kuleshov and Rudakov's t-stabilities generalise Bridgeland's theory of stability conditions. As an application of this notion, we use a complete classification of 'generalised' co-slicings in the bounded derived category of the Kronecker algebra, $D^b(KQ)$, to obtain a classification of co-t-structures in $D^b(Q)$. This is then used to compute the co-stability manifold of $D^b(KQ)$.

Coloured quivers for rigid objects and partial triangulations: The unpunctured case

We associate a coloured quiver to a rigid object in a Hom-finite 2-Calabi--Yau triangulated category and to a partial triangulation on a marked (unpunctured) Riemann surface. We show that, in the case where the category is the generalised cluster category associated to a surface, the coloured quivers coincide. We also show that compatible notions of mutation can be defined and give an explicit description in the case of a disk. A partial description is given in the general 2-Calabi-Yau case. We show further that Iyama-Yoshino reduction can be interpreted as cutting along an arc in the surface.