Related papers: Reflexivity and Hochschild Cohomology
We study the classification of submodules of module categories over monoidal categories, extending ideas of Coulembier on the classification of tensor ideals in monoidal categories. We develop a framework that applies to module categories…
We continue our study of the monoidal category $D(G)$. At the level of cohomology we transfer the duality functor to the derived category of Hecke dg-modules. In the process we develop a more general and streamlined approach to the…
Two unital operator algebras A, B are called Delta-equivalent if there exists an equivalence functor between the categories A-mod and B-mod which "extends" to a *-functor implementing an equivalence between the categories A-dmod and B-dmod.…
We present a bicategorical perspective on derived Morita theory for rings, DG algebras, and spectra. This perspective draws a connection between Morita theory and the bicategorical Yoneda Lemma, yielding a conceptual unification of Morita…
We associate a monoidal category $\mathcal{H}^\lambda$ to each dominant integral weight $\lambda$ of $\widehat{\mathfrak{sl}}_p$ or $\mathfrak{sl}_\infty$. These categories, defined in terms of planar diagrams, act naturally on categories…
In this work, we study the Hochschild-Mitchell Cohomology of triangular matrix categories. Given a triangular matrix category $\Lambda=\left[ \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$, we investigate…
We prove that for a self-injective ribbon Grothendieck-Verdier category $\mathcal{C}$ in the sense of Boyarchenko-Drinfeld the cyclic action on the Hochschild complex of $\mathcal{C}$ extends to an action of the diffeomorphism group of the…
An adjoint pair of contravariant functors between abelian categories can be extended to the adjoint pair of their derived functors in the associated derived categories. We describe the reflexive complexes and interpret the achieved results…
Linear categories naturally have several identification relations : isomorphisms, categorical equivalences and Morita equivalences. In this thesis, we construct the classifying stacks for these three relations ($\ukcatiso$, $\ukcateq$,…
We investigate the homological behaviour of compactly generated triangulated categories under separable extensions. We show that homological invariants (finiteness of global dimension, gorensteinness and regularity) are preserved under such…
In this article we study the cohomology of deep level Deligne--Lusztig varieties of Coxeter type, attached to a reductive group over a local non-archimedean field, which splits over an unramified extension. This allows to construct some new…
A cocycle category H(X,Y) is defined for objects X and Y in a model category, and it is shown that the set of morphisms [X,Y] is isomorphic to the set of path components of H(X,Y) provided the ambient model category is right proper and…
We describe the (bigraded) Hochschild cohomology of graded gentle algebras along with the Gerstenhaber bracket and cup product. In particular, this yields a description of the Hochschild cohomology of partially wrapped Fukaya categories of…
For every imprimitive complex reflection group of rank 2, we construct a semi-orthogonal decomposition of the derived category of the associated global quotient stack which categorifies the usual decomposition of the orbifold cohomology…
In this paper, we construct an explicit minimal projective bimodule resolution of a self-injective special biserial algebra $A_{T}$ ($T\geq0$) whose Grothendieck group is of rank $4$. As a main result, we determine the dimension of the…
Lazard correspondence provides an isomorphism of categories between finitely generated nilpotent pro-$p$ groups of nilpotency class smaller than $p$ and finitely generated nilpotent $\mathbb{Z}_p$-Lie algebras of nilpotency class smaller…
We present a criterion for establishing Morita equivalence of monoidal categories, and apply it to the categorical representation theory of reductive groups $G$. We show that the "de Rham group algebra" $\mathcal D(G)$ (the monoidal…
We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the…
We define (iterated) coisotropic correspondences between derived Poisson stacks, and construct symmetric monoidal higher categories of derived Poisson stacks where the $i$-morphisms are given by $i$-fold coisotropic correspondences.…
Given any modular category $\mathcal{C}$ over an algebraically closed field $k$, we extract a sequence $(M_g)_{g\geq 0}$ of $\mathcal{C}$-bimodules. We show that the Hochschild chain complex $CH(\mathcal{C};M_g)$ of $\mathcal{C}$ with…