The goal of this paper is to study the homotopy theory of homotopy algebras over a Koszul operad with their infinity morphisms. The method consists in endowing the category of coalgebras over the Koszul dual cooperad with a model category structure. CONTENTS Introduction 1 1. Recollections 2 2. Model category structure for coalgebras 5 3. Conclusion 14 Appendix A. A technical lemma 15 Reference...

Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9],[19], [13], [24],[15], [18], [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] bar construction of A. We show that these methods are equivalent and are determined combinatorially...

The goal of this note is to prove that the category of perverse sheaves constructible with respect to a triangulation is Koszul (i.e. equivalent to the category of finite-dimensional representations of a Koszul algebra). This result was conjectured by M. Vybornov in the framework of the general theory of cellular perverse sheaves due to R. MacPherson [4], [5]. Acknowledgments. I am grateful to ...

We give a complete picture of the interaction between Koszul and Ringel dualities for quasi-hereditary algebras admitting linear tilting (co)resolutions of standard and costandard modules. We show that such algebras are Koszul, that the class of these algebras is closed with respect to both dualities and that on this class these two dualities commute. All arguments reduce to short computations ...

In this paper we prove that the coordinate ring of the pinched Veronese (i.e k[X, XY, XY , Y , XZ, Y Z, XZ, Y Z, Z] ⊂ k[X, Y, Z]) is Koszul. The strategy of the proof is the following: we can consider a presentation S/I where S = k[X1, . . . , X9]. Using a distinguished weight ω, it’s enough to show that S/inωI is Koszul. We write inωI as J + H where J is generated by a Gröbner basis of quadric...

We are going to show that the sheafication of graded Koszul modules KΓ over Γn = K [x0, x1...xn] form an important subcategory ∧ KΓ of the coherents sheaves on projective space, Coh(P n). One reason is that any coherent sheave over P n belongs to ∧ KΓup to shift. More importantly, the category KΓ allows a concept of almost split sequence obtained by exploiting Koszul duality between graded Kosz...

There are many structures (algebras, categories, etc) with natural gradings such that the degree 0 components are not semisimple. Particular examples include tensor algebras with non-semisimple degree 0 parts, extension algebras of standard modules of standardly stratified algebras. In this thesis we develop a generalized Koszul theory for graded algebras (categories) whose degree 0 parts may b...

This paper studies the homological determinants and Nakayama automorphisms of not-necessarily-noetherian m-Koszul twisted Calabi-Yau or, equivalently, m-Koszul Artin-Schelter regular, algebras. Dubois-Violette showed that such an algebra is isomorphic to a derivation quotient algebra D(w, i) for a unique-up-to-scalar-multiples twisted superpotential w. By definition, D(w, i) is the quotient of ...

A cubical Feynman category, introduced by the authors in previous work, is a category whose functors to base C <mml:annotation encoding="appl...