The category of modules over a monoidal category: Abelian or not?

LetA be an algebra in an abelian monoidal category M. We prove that the category of leftA-modules is abelian, wheneverA is right coflat. This paper was written while the author was member of G.N.S.A.G.A. with partial financial support from M.I.U.R.     

Classifying spaces for braided monoidal categories and lax diagrams of bicategories

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well-understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be thought of as the homotopy colimit of the classifying spaces of the bicategories that arise from the initial input data given by the lax diagram. This result is applied to produce bicategories whose classifying space has a double loop space with the same homotopy type, up to group completion, as the underlying category of any given (non-necessarily strict) braided monoidal category. Specifically, it is proven that these double delooping spaces, for categories enriched with a braided monoidal structure, can be explicitly realized by means of certain genuine simplicial sets characteristically associated to any braided monoidal categories, which we refer to as their (Street's) geometric nerves.     

Hopf monads on monoidal categories

We define Hopf monads on an arbitrary monoidal category, extending the definition given in Bruguières and Virelizier (2007) [5] for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode.Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a category of bimodules admitting a right adjoint). We show that any finite tensor category is the category of finite-dimensional modules over a Hopf algebroid.Any Hopf algebra in the center of a monoidal category C gives rise to a Hopf monad on C. The Hopf monads so obtained are exactly the augmented Hopf monads. More generally if a Hopf monad T is a retract of a Hopf monad P, then P is a cross product of T by a Hopf algebra of the center of the category of T-modules (generalizing the Radford–Majid bosonization of Hopf algebras).We show that the comonoidal comonad of a Hopf adjunction is canonically represented by a cocommutative central coalgebra. As a corollary, we obtain an extension of Sweedler?s Hopf module decomposition theorem to Hopf monads (in fact to the weaker notion of pre-Hopf monad).     

The relative Picard functor on schemes over a symmetric monoidal category

We consider schemes (X,OX) over an abelian closed symmetric monoidal category (C,⊗,1). Our aim is to extend a theorem of Kleiman on the relative Picard functor to schemes over (C,⊗,1). For this purpose, we also develop some basic theory on quasi-coherent modules on schemes (X,OX) over C.     

Gorenstein projective bimodules via monomorphism categories and filtration categories

We generalize the monomorphism category from quiver (with monomial relations) to arbitrary finite dimensional algebras by a homological definition. Given two finite dimension algebras A and B, we use the special monomorphism category $\mathsf{Mon}(B,A\text{-}\mathsf{Gproj})$ to describe some Gorenstein projective bimodules over the tensor product of A and B. If one of the two algebras is Gorenstein, we give a sufficient and necessary condition for $\mathsf{Mon}(B,A\text{-}\mathsf{Gproj})$ being the category of all Gorenstein projective bimodules. In addition, if both A and B are Gorenstein, we can describe the category of all Gorenstein projective bimodules via filtration categories. Similarly, in this case, we get the same result for infinitely generated Gorenstein projective bimodules.     

On free abelian categories for theorem proving

We give a computational approach to theorem proving in homological algebra. This approach is based on computations in the free abelian category of an additive category A. We show that the free abelian category is amenable to explicit computations whenever we can decide homotopy equations in A. As some consequences of our investigations, we recover Dowker's explicit formula for the connecting homomorphism ? in the snake lemma, we find a universal sense in which ? is unique, and we give a refined version of the 5-lemma.     

Traces in symmetric monoidal categories

The purpose of this expository note is to describe duality and trace in a symmetric monoidal category, along with important properties (including naturality and functoriality), and to give as many examples as possible. Among other things, this note is intended as background for the generalizations to the context of bicategories and indexed monoidal categories.     

Exact categories,big Cohen-Macaulay modules and finite representation type

One of the first remarkable results in the representation theory of artin algebras, due to Auslander and Ringel-Tachikawa, is the characterisation of when an artin algebra is representation-finite. In this paper, we investigate aspects of representation-finiteness in the general context of exact categories in the sense of Quillen. In this framework, we introduce “big objects” and prove an Auslander-type “splitting-big-objects” theorem. Our approach generalises and unifies the known results from the literature. As a further application of our methods, we extend the theorems of Auslander and Ringel-Tachikawa to arbitrary dimension, i.e. we characterise when a Cohen-Macaulay order over a complete regular local ring is of finite representation type.     

Invertible bimodule categories over the representation category of a Hopf algebra

For any finite-dimensional Hopf algebra H   we construct a group homomorphism BiGal(H)→BrPic(Rep(H))$\mathrm{BiGal}(H)\to \text{BrPic}(\mathrm{Rep}(H))$, from the group of equivalence classes of H  -biGalois objects to the group of equivalence classes of invertible exact Rep(H)$\mathrm{Rep}(H)$-bimodule categories. We discuss the injectivity of this map. We exemplify in the case H=_Tq$H=T_q$ is a Taft Hopf algebra and for this we classify all exact indecomposable Rep(_Tq)$\mathrm{Rep}(T_q)$-bimodule categories.     

Classification of split torsion torsionfree triples in module categories

A TTF-triple (C,T,F) in an abelian category is one-sided split in case either (C,T) or (T,F) is a split torsion theory. In this paper we classify one-sided split TTF-triples in module categories, thus completing Jans’ classification of two-sided split TTF-triples and answering a question that has remained open for almost 40 years.     

Perpendicular categories of infinite dimensional partial tilting modules and transfers of tilting torsion classes

Let R be a ring and P be an (infinite dimensional) partial tilting module. We show that the perpendicular category of P is equivalent to the full module category where and ?R is the Bongartz complement of P modulo its P-trace. Moreover, there is a ring epimorphism φ:R→S. We characterize the case when φ is a perfect localization. By [Riccardo Colpi, Alberto Tonolo, Jan Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 (10) (1997) 3225-3237], there exist mutually inverse isomorphisms μ^′ and ν^′ between the interval in the lattice of torsion classes in , and the lattice of all torsion classes in . We provide necessary and sufficient conditions for μ^′ and ν^′ to preserve tilting torsion classes. As a consequence, we show that these conditions are always satisfied when R is a Dedekind domain, and if P is finitely presented and R is an artin algebra, then the conditions reduce to the trivial ones, namely that each value of μ^′ and ν^′ contains all injectives.     

Expansions of abelian categories

Expansions of abelian categories are introduced. These are certain functors between abelian categories and provide a tool for induction/reduction arguments. Expansions arise naturally in the study of coherent sheaves on weighted projective lines; this is illustrated by various applications.     

Liftings of distributive lattices by locally matricial algebras with respect to the Id c functor

We study representations of distributive -lattices, considered as join-semilattices, by semilattices of finitely generated two-sided ideals of locally matricial algebras over a field k, aiming to find a functorial solution of the problem. We find simple examples of a finite subcategory of the category Ld of distributive -lattices and of a subcategory of L_d corresponding to a partially ordered class which cannot be lifted with respect to the Id_c functor. On the other hand, we prove that there is such a lifting of every diagram in L_d or of a subcategory L_d1 of L_d whose objects are all distributive -lattices and whose morphisms are -embeddings. This paper is dedicated to Walter Taylor. Received February 8, 2005; accepted in final form August 11, 2005. The work is a part of the research project MSM 0021620839 financed by MSMT and partly supported by INTAS project 03-51-4110, the grant GAUK 448/2004/B-MAT, and the post-doctoral grant GAČR 201/03/P140.     

Structure sheaves of definable additive categories

We prove that the 2-category of small abelian categories with exact functors is anti-equivalent to the 2-category of definable additive categories. We define and compare sheaves of localisations associated to the objects of these categories. We investigate the natural image of the free abelian category over a ring in the module category over that ring and use this to describe a basis for the Ziegler topology on injectives; the last can be viewed model-theoretically as an elimination of imaginaries result.     

Finitely semisimple spherical categories and modular categories are self-dual

We show that every essentially small finitely semisimple k-linear additive spherical category for which k=End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category, with respect to the long forgetful functor, is self-dual as a Weak Hopf Algebra.     

On the (non)vanishing of some “derived” categories of curved dg algebras

Since curved dg algebras, and modules over them, have differentials whose square is not zero, these objects have no cohomology, and there is no classical derived category. For different purposes, different notions of “derived” categories have been introduced in the literature. In this article, we show that for some concrete curved dg algebras, these derived categories vanish. This happens for example for the initial curved dg algebra whose module category is the category of precomplexes, and for certain deformations of dg algebras.