On equivariant triangulated categories

Abstract

Consider a finite group $G$ acting on a triangulated category $\mathcal{T}$. In this paper we investigate triangulated structure on the category ${\mathcal{T}}^G$ of $G$-equivariant objects in $\mathcal{T}$. We prove (under some technical conditions) that such structure exists. Supposed that an action on $\mathcal{T}$ is induced by a DG-action on some DG-enhancement of $\mathcal{T}$, we construct a DG-enhancement of ${\mathcal{T}}^G$. Also, we show that the relation "to be an equivariant category with respect to a finite abelian group action" is symmetric on idempotent complete additive categories.

28 pages. Remarks are welcome. v2: Our point of view has changed and the paper is reorganized respectively. A chapter about existence of triangulation on ${\mathcal{T}}^G$ is added

Cited by in corpus (14)

• Equivalences of equivariant derived categories
• The Space of Stability Conditions on Abelian Threefolds, and on some Calabi-Yau Threefolds
• Group actions on categories and Elagin's Theorem Revisited
• P-functor versions of the Nakajima operators
• Universal functors on symmetric quotient stacks of Abelian varieties
• Equivariant split generation and mirror symmetry of special isogenous tori
• A note on equivariantization of additive categories and triangulated categories
• Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles
• Rational points on symmetric powers and categorical representability
• Admissible homomorphisms and equivariant relations between weighted projective lines
• A power structure over the Grothendieck ring of geometric dg categories
• The dual actions, equivariant autoequivalences and stable tilting objects
• Weighted projective lines of tubular type and equivariantization
• CPM Categories for Galois Extensions