K-linear Category Research Articles

Serre Functors and Graded Categories

We study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a graded Nakayama automorphism of finite order. This connects various results in the literature and gives new examples of fractional Calabi-Yau algebras.

Cartesian Differential Categories as Skew Enriched Categories

We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

Higher Dualizability and Singly-Generated Grothendieck Categories

Let k be a field. We show that locally presentable, k-linear categories $${\mathcal {C}}$$ dualizable in the sense that the identity functor can be recovered as $$\coprod _i x_i\otimes f_i$$ for objects $$x_i\in {\mathcal {C}}$$ and left adjoints $$f_i$$ from $${\mathcal {C}}$$ to $$\mathrm {Vect}_k$$ are products of copies of $$\mathrm {Vect}_k$$ . This partially confirms a conjecture by Brandenburg, the author and T. Johnson-Freyd. Motivated by this, we also characterize the Grothendieck categories containing an object x with the property that every object is a copower of x: they are precisely the categories of non-singular injective right modules over simple, regular, right self-injective rings of type I or III.

Entwined modules over linear categories and Galois extensions

In this paper, we study modules over quotient spaces of certain categorified fiber bundles. These are understood as modules over entwining structures involving a small K-linear category $${\cal D}$$ and a K-coalgebra C. We obtain Frobenius and separability conditions for functors on entwined modules. We also introduce the notion of a C-Galois extension $${\cal E} \subseteq {\cal D}$$ of categories. Under suitable conditions, we show that entwined modules over a C-Galois extension may be described as modules over the subcategory $${\cal E}$$ of C-coinvariants of $${\cal D}$$ .

D-Representation-finite self-injective algebras

In this paper, we initiate the study of higher-dimensional Auslander–Reiten theory of self-injective algebras. We give a systematic construction of (weakly) d-representation-finite self-injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the Serre functor. The condition holds, in particular, for all fractionally Calabi-Yau algebras of global dimension at most d. This generalizes Riedtmann's classical construction of representation-finite self-injective algebras. Our method is based on an adaptation of Gabriel's covering theory for k-linear categories to the setting of higher-dimensional Auslander–Reiten theory.Applications include n-fold trivial extensions and (classical and higher) preprojective algebras, which are shown to be d-representation-finite in many cases. We also get a complete classification of all d-representation-finite self-injective Nakayama algebras for arbitrary d.

A remark on a theorem by Claire Amiot

Claire Amiot has classified the connected triangulated k-categories with finitely many isoclasses of indecomposables satisfying suitable hypotheses. We remark that her proof shows that these triangulated categories are determined by their underlying k-linear categories. We observe that, if the connectedness assumption is dropped, the triangulated categories are still determined by their underlying k-categories together with the action of the suspension functor on the set of isoclasses of indecomposables.

On Standard Derived Equivalences of Orbit Categories

Let k be a commutative ring, $\mathcal {A}$ and $\mathcal {B}$ – two k-linear categories with an action of a group G. We introduce the notion of a standard G-equivalence from $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$ to $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ , where $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ is the homotopy category of finitely generated projective $\mathcal {A}$ -complexes. We construct a map from the set of standard G-equivalences to the set of standard equivalences from $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$ to $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ and a map from the set of standard G-equivalences from $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$ to $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ to the set of standard equivalences from $\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {B}/G)$ to $\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {A}/G)$ , where $\mathcal {A}/G$ denotes the orbit category. We investigate the properties of these maps and apply our results to the case where $\mathcal {A}=\mathcal {B}=R$ is a Frobenius k-algebra and G is the cyclic group generated by its Nakayama automorphism ν. We apply this technique to obtain the generating set of the derived Picard group of a Frobenius Nakayama algebra over an algebraically closed field.

Blocks of Mackey categories

For a suitable small category F of homomorphisms between finite groups, we introduce two subcategories of the biset category, namely, the deflation Mackey category MF← and the inflation Mackey category MF→. Let G be the subcategory of F consisting of the injective homomorphisms. We shall show that, for a field K of characteristic zero, the K-linear category KMG=KMG←=KMG→ has a semisimplicity property and, in particular, every block of KMG owns a unique simple functor up to isomorphism. On the other hand, we shall show that, when F is equivalent to the category of finite groups, the K-linear categories KMF← and KMF→ each have a unique block.

Stratifications of Finite Directed Categories and Generalized APR Tilting Modules

A finite directed category is a k-linear category with finitely many objects and an underlying poset structure, where k is an algebraically closed field. This concept unifies structures such as k-linerizations of posets and finite EI categories, quotient algebras of finite-dimensional hereditary algebras, triangular matrix algebras, etc. In this article, we study representations of finite directed categories and discuss their stratification properties. In particular, we show the existence of generalized Auslander-Platzeck-Reiten tilting modules for triangular matrix algebras under some assumptions.

Partial Hopf module categories

The effectiveness of the application of constructions in G-graded k-categories to the computation of the fundamental group of a finite dimensional k-algebra, alongside with open problems still left untouched by those methods and new problems arisen from the introduction of the concept of fundamental group of a k-linear category, motivated the investigation of H-module categories, i.e., actions of a Hopf algebra H on a k-linear category. The G-graded case corresponds then to actions of the Hopf algebra kG on a k-linear category, where kG is the dual group algebra of G. In this work we take a step further and introduce partial H-module categories. We extend several results of partial H-module algebras to this context, such as the globalization theorem, the construction of the partial smash product and the Morita equivalence of this category with the smash product over a globalization. We also present a detailed description of partial actions of kG.

On the regular representation of an (essentially) finite 2-group

The regular representation of an essentially finite 2-group G in the 2-category 2Vect k of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it are computed. It is next shown that all hom-categories in Rep 2Vect k ( G ) are 2-vector spaces under quite standard assumptions on the field k, and a formula giving the corresponding “intertwining numbers” is obtained which proves they are symmetric. Finally, it is shown that the forgetful 2-functor ω : Rep 2Vect k ( G ) → 2Vect k is representable with the regular representation as representing object. As a consequence we obtain a k-linear equivalence between the 2-vector space V ect k G of functors from the underlying groupoid of G to V ect k , on the one hand, and the k-linear category E nd ( ω ) of pseudonatural endomorphisms of ω , on the other hand. We conclude that E nd ( ω ) is a 2-vector space, and we (partially) describe a basis of it.

Separable K-linear categories

Abstract We define and investigate separable K-linear categories. We show that such a category C is locally finite and that every left C-module is projective. We apply our main results to characterize separable linear categories that are spanned by groupoids or delta categories.

HOCHSCHILD–MITCHELL COHOMOLOGY OF A LOCALLY BOUNDED CATEGORY

We consider k-linear categories A which are locally bounded, that is, for every object x, the sums ∑y dim k A(x, y) and ∑y dim k A(y, x) are finite. Let x1,x2,…, be an enumeration of the objects of A such that for each n ∈ ℕ, the full subcategory A(n) of A formed by the objects x1,x2,…,xn is convex in A. We compute the Hochschild–Mitchell cohomology of A as the inverse limit of the cohomologies of the categories A(n).

Derived invariance of Hochschild–Mitchell (co)homology and one-point extensions

In this article we prove derived invariance of Hochschild–Mitchell homology and cohomology and we extend to k-linear categories a result by Barot and Lenzing concerning derived equivalences and one-point extensions. We also prove the existence of a long exact sequence à la Happel and we give a generalization of this result which provides an alternative approach.

2-Cosemisimplicial objects in a 2-category, permutohedra and deformations of pseudofunctors

In this paper we take up again the deformation theory for K-linear pseudofunctors initiated in Elgueta (Adv. Math. 182 (2004) 204–277). We start by introducing a notion of 2-cosemisimplicial object in an arbitrary 2-category and analyzing the corresponding coherence question, where the permutohedra make their appearance. We then describe a general method to obtain usual cochain complexes of K-modules from (enhanced) 2-cosemisimplicial objects in the 2-category Cat K of small K-linear categories and prove that the deformation complex X •( F) introduced in Elgueta (to appear) can be obtained by this method from a 2-cosemisimplicial object that can be associated to F . Finally, using this 2-cosemisimplicial object of F and a generalization to the context of K-linear categories of the deviation calculus introduced by Markl and Stasheff for K-modules (J. Algebra 170 (1994) 122), it is shown that the obstructions to the integrability of an nth-order deformation of F indeed correspond to cocycles in the third cohomology group H 3(X •( F)) , a question which remained open in Elgueta (Adv. Math. 182 (2004) 204–277).

Modular functors in homotopy quantum field theory and tortile structures

We study a version of a modular functor for Turaev's homotopy quantum field theories using 2-categories of surfaces. We define the homotopy surface 2-category of a space X and define an S X -structure to be a monoidal 2-functor from this to the 2-category of idempotent-complete additive k-linear categories. We initiate the study of the algebraic structure arising from these functors. In particular we show that a unitary S X -structure gives rise to a lax tortile π-category when the background space is an Eilenberg–Maclane space X= K( π,1), and to a tortile category with lax π 2 X-action when the background space is simply connected.

Chern maps for U-theory

For an exact k-linear category A with a duality functor, the dihedral homology of A is defined. We show that when 1 2 ϵ k , the image of the generalized Chern map from the U-theory of A to cyclic homology, lies in a direct summand which is the dihedral homology of A . When A is the category of finitely generated projective A-modules over a k-algebra A, we recover the early results of Cortinas (1993) and Lodder (1992). The approach we adopt here is more categorical, inspired by McCarthy's work (1992).

Embedding rings in completed graded rings 1. Triangular embeddings

Let k be a commutative ring with 1, R an associative k-algebra without 1, and n a positive integer. It is shown that the following conditions are equivalent: (a) R can be embedded as a k-algebra in the ring of strictly upper triangular n × n matrices over an associative k-algebra A. (b) R has such an embedding with A further assumed commutative, (c) R can be embedded as a k-algebra in a graded associative k-algebra C whose only nonzero components are C 1,…, C nn − 1 . The proof that the other conditions imply (b) is essentially a demonstration that a certain kind of k-linear category always has a faithful functor into a commutative k-algebra. Subsequent papers in this series determine sufficient conditions for (a)–(c) above to hold, in terms of nilpotence of R, under various hypotheses