Calabi-Yau (CY) triangulated categories are those satisfying a useful and important
duality, characterised by a number called the CY dimension. Much work
has been carried out on understanding positive CY triangulated categories, especially in the
context of cluster-tilting theory. Even though CY dimension is usually
considered to be a positive (or fractional) number, there are natural
examples of CY triangulated categories where this”dimension” or parameter is negative,
for example, stable module categories of self-injective algebras. Therefore, negative CY
triangulated categories constitute a class of categories that warrant
further systematic study. In this talk, we will give a brief
survey regarding what is so far known about the structure of negative
CY triangulated categories, and we will focus on an example given by
triangulated categories generated by spherical objects.