In this paper we study Schlichting's K-theory groups of the Buchweitz-Orlov singularity category $\mathcal{D}^{sg}(X)$ a quasi-projective algebraic scheme $X/k$ with applications to Algebraic K-theory. We prove that for isolated quotient singularities $\mathrm{K}_0(\mathcal{D}^{sg}(X))$ is finite torsion, and $\mathrm{K}_1(\mathcal{D}^{sg}(X)) = 0$. One main varieties satisfy rational Poincare ...

Given a finite tensor category $\mathcal {C}$ , we prove that modified trace on the ideal of projective objects can be obtained from suitable trivialization Nakayama functor as right -module functor. Using result Costello, this allows us to associate any equipped with such chain complex valued topological conformal field theory, theory. The theory topologically describes trace, Hattori-Stalling...

In the past recent years, I have been using string diagrams to teach basic category theory (adjunctions, Kan extensions, but also limits and Yoneda embedding). Using graphical notations is undoubtedly joyful, and brings us close to other graphical syntaxes of circuits, interaction nets, etc... It saves us from laborious verifications of naturality, which is built-in in string diagrams. On the o...

1 Network System Research Institute, National Institute of Information and Communications Technology, 4-2-1 Nukui-kita, Koganei, Tokyo 184-8795, Japan 2 WPI Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan 3 Earth-Life Science Institute, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan...