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We fill a gap in the literature regarding `transport of structure' for (n+2)-angulated, n-exact, n-abelian and n-exangulated categories appearing in (classical and higher) homological algebra. As an application of our main results, we show that a skeleton of one of these kinds of categories inherits the same structure in a canonical way, up to equivalence. In particular, it follows that a skeleton of a weak (n+2)-angulated category is in fact what we call a strong (n+2)-angulated category. When n=1 this clarifies a technical concern with the definition of a cluster category. We also introduce the notion of an n-exangulated functor between n-exangulated categories. This recovers the definition of an (n+2)-angulated functor when the categories concerned are (n+2)-angulated, and the higher analogue of an exact functor when the categories concerned are n-exact., Comment: v3: 24 pages; minor typographical changes; accepted in Journal of Algebra

Suppose $$F:{\mathcal {D}}(X)\rightarrow {\mathcal {T}}$$ is an exact functor from the bounded derived category of coherent sheaves on a smooth projective variety X to a triangulated category $${\mathcal {T}}$$ . If F possesses left and right adjoints, then the Bondal–Orlov criterion gives a simple way of determining if F is fully faithful. We prove a natural extension of this theorem to the case when X is a smooth and proper DM stack with projective coarse moduli space.

In the previous article "Hearts of twin cotorsion pairs on exact categories", we introduced the notion of the heart for any cotorsion pair on an exact category with enough projectives and injectives, and showed that it is an abelian category. In this paper, we construct a half exact functor from the exact category to the heart. This is analog of the construction of Abe and Nakaoka for triangulated categories. We will also use this half exact functor to find out a sufficient condition when two different hearts are equivalent., 26 pages

Let W m | n be the (finite) W-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra gl m | n ( C ) . In this paper we study the Whittaker coinvariants functor, which is an exact functor from category O for gl m | n ( C ) to a certain category of finite-dimensional modules over W m | n . We show that this functor has properties similar to Soergel's functor V in the setting of category O for a semisimple Lie algebra. We also use it to compute the center of W m | n explicitly, and deduce consequences for the classification of blocks of O up to Morita/derived equivalence.

We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor $\Sigma$ on a finite abelian category $\mathcal{M}$, we introduce the notion of a $\Sigma$-twisted trace on the class $\mathrm{Proj}(\mathcal{M})$ of projective objects of $\mathcal{M}$. In our framework, there is a one-to-one correspondence between the set of $\Sigma$-twisted traces on $\mathrm{Proj}(\mathcal{M})$ and the set of natural transformations from $\Sigma$ to the Nakayama functor of $\mathcal{M}$. Non-degeneracy and compatibility with the module structure (when $\mathcal{M}$ is a module category over a finite tensor category) of a $\Sigma$-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular., Comment: 39 pages; to appear in Algebras and Representation Theory

We study relative $K_0$ of exact categories and triangulated categories. As an application, we construct a cycle class map from Chow groups with modulus to relative $K_0$., 20 pages, final version, to appear in Journal of Pure and Applied Algebra

For any ring R, the Auslander–Gruson–Jensen functor D A : fp ( R - Mod , Ab ) → ( mod - R , Ab ) op is the exact functor which sends a representable functor ( X , − ) to the tensor functor − ⊗ X . We show that this functor admits a fully faithful right adjoint D R and a fully faithful left adjoint D L . That is, we show that D A is part of a recollement of abelian categories. In particular, this shows that D A is a localisation and a colocalisation which gives an equivalence of categories fp ( R - Mod , Ab ) { F : D A F = 0 } ≃ ( mod - R , Ab ) op . We show that { F : D A F = 0 } is the Serre subcategory of fp ( R - Mod , Ab ) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R - Mod → Ab is also finitely presented.

Given a higher-rank graph $\Lambda$, we investigate the relationship between the cohomology of $\Lambda$ and the cohomology of the associated groupoid $G_\Lambda$. We define an exact functor between the abelian category of right modules over a higher-rank graph $\Lambda$ and the category of $G_\Lambda$-sheaves, where $G_\Lambda$ is the path groupoid of $\Lambda$. We use this functor to construct compatible homomorphisms from both the cohomology of $\Lambda$ with coefficients in a right $\Lambda$-module, and the continuous cocycle cohomology of $G_\Lambda$ with values in the corresponding $G_\Lambda$-sheaf, into the sheaf cohomology of $G_\Lambda$., Comment: A flaw in the proof of Proposition 4.2 in v1 of this paper has invalidated Proposition 4.8 and Theorem 4.9 from v1. This version (v3) has been substantially revised and includes new results. Version 4 to appear in Banach J. Math. Anal

Making use of Freyd's free abelian category on a preadditive category we show that if $T:D\rightarrow \mathcal{A}$ is a representation of a quiver $D$ in an abelian category $\mathcal{A}$ then there is an abelian category $\mathcal{A} (T)$, a faithful exact functor $F_T: \mathcal{A} (T) \to \mathcal{A}$ and an induced representation $\tilde T: D \to \mathcal{A} (T)$ such that $F_T\tilde T= T$ universally. We then can show that $\mathbb{T}$-motives as well as Nori's motives are given by a certain category of functors on definable categories.

This paper presents a sound and complete sequent-style deduction system for determining behavioural equivalence in coalgebras of finitary set functors preserving weak pullbacks. We also prove soundness without the weak pullback requirement. Finitary set functors are investigated because they are quotients of polynomial functors: the polynomial functor provides a ready-made signature and the quotient provides necessary additional axioms. We also show that certain operations on functors can be expressed with uniform changes to the presentations of the input functors, making this system compositional for a range of widely-studied classes of functors, including the Kripke polynomial functors. Our system has roots in the F L R 0 proof system of Moschovakis et al., particularly as used by Moss, Wennstrom, and Whitney for non-wellfounded sets. Similarities can also be drawn to expression calculi in the style of Bonsangue, Rutten, and Silva.

We generalize a result of Orlov and Van den Bergh on the representability of a cohomological functor from the bounded derived category of a smooth projective variety over a field to the category of L-modules, to the case where L is a field extension of the base field k of the variety X, with L of transcendence degree less than or equal to one or L purely transcendental of degree 2. This result can be applied to investigate the behavior of an exact functor between the bounded derived categories of coherent sheaves of X and Y, with X and Y smooth projective and Y of dimension less than or equal to one or Y a rational surface. We show that for any such F there exists a "generic kernel" A in the derived category of the product, such that F is isomorphic to the Fourier-Mukai transform with kernel A after composing both with the pullback to the generic point of Y., Comment: to appear in Journal of Noncommutative Geometry

An additive functor [Formula: see text] → [Formula: see text] between additive categories is objective if any morphism f in [Formula: see text] with F(f) = 0 factors through an object K with F(K) = 0. We consider when a triangle functor in an adjoint pair is objective. We show that a triangle functor is objective provided that its adjoint (whatever left adjoint or right adjoint) is full or dense. We also give an example to show that the adjoint of a faithful triangle functor is not necessarily objective. In particular, the adjoint of an objective triangle functor is not necessarily objective. This is in contrast to the well-known fact that the adjoint of a triangle functor is always a triangle functor. Also, for an arbitrary adjoint pair (F, G) between categories which are not necessarily additive, we give a sufficient and necessary condition such that F (resp., G) is full or faithful.

We lift to equivariant algebra three closely related classical algebraic concepts: abelian group objects in augmented commutative algebras, derivations, and Kahler differentials. We define Mackey functor objects in the category of Tambara functors augmented to a fixed Tambara functor R _ , and we show that the usual square-zero extension gives an equivalence of categories between these Mackey functor objects and ordinary modules over R _ . We then describe the natural generalization to Tambara functors of a derivation, building on the intuition that a Tambara functor has products twisted by arbitrary finite G-sets, and we connect this to square-zero extensions in the expected way. Finally, we show that there is an appropriate form of Kahler differentials which satisfy the classical relation that derivations out of R _ are the same as maps out of the Kahler differentials.

In this paper, we show that the higher derived functors of the primitive element functor of $$E_*(M)$$ where M is a quasitoric manifold and E is a complex-orientable homology theory are independent of the torus action, implying they depend on the orbit space. This extends the results of an earlier work by both authors. By applying cosimplicial methods to coalgebras that are free modules over a commutative ring, we are able to generalize the aforementioned functors that appeared in Bousfield’s work. In addition, certain results that appeared in earlier work of Larry Smith have been generalized and applied to faithful systems to expose how and to what extent ESP sequences show up in the arguments. As an application, we are able to prove that a necessary condition for a simplicial complex dual to a simple convex polytope being rigid is an isomorphism of these derived functors in sufficiently large dimensions; this further exposes the relation between the homotopy type of the manifold, the torus action and the combinatorics of the orbit.

In the work we investigate categorical and topological properties of the functor O S τ of semiadditive τ-smooth functionals in the category T y c h of Tychonoff spaces and their continuous mappings, which extends the functor OS of semiadditive functionals in the category C o m p of compact spaces and their continuous mappings. It is proved that the functor O S τ is a normal functor in the category T y c h of Tychonoff spaces.

In this paper, we investigate the homological properties of the functor categories (mod−R, Ab) and ((mod−R)op, Ab). Some new homological dimensions in these functor categories such as FP-projecive dimensions and cotorsion dimensions for functors and functor categories are introduced and studied. We also characterize functor categories of homological dimensions zero and explore the connections among some different homological dimensions.

Rickard proved that for certain self-injective algebras, a stable equivalence induced from an exact functor is a stable equivalence of Morita type, in the sense of Brou\'{e}. In this paper we study singular equivalences of finite dimensional algebras induced from tensor product functors. We prove that for certain Gorenstein algebras, a singular equivalence induced from tensoring with a suitable complex of bimodules, induces a singular equivalence of Morita type with level, in the sense of Wang. This recovers Rickard's theorem in the self-injective case., Comment: Final version. The proof of the main Corollary is now fixed as well as some typos in 3.6, 3.7. To appear in the Bulletin of the London Mathematical Society

Let $T$ be a right exact functor from an abelian category $\mathscr{B}$ into another abelian category $\mathscr{A}$. Then there exists a functor ${\bf p}$ from the product category $\mathscr{A}\times\mathscr{B}$ to the comma category $(T\downarrow\mathscr{A})$. In this paper, we study the property of the extension closure of some classes of objects in $(T\downarrow\mathscr{A})$, the exactness of the functor ${\bf p}$ and the detail description of orthogonal classes of a given class ${\bf p}(\mathcal{X},\mathcal{Y})$ in $(T\downarrow\mathscr{A})$. Moreover, we characterize when special precovering classes in abelian categories $\mathscr{A}$ and $\mathscr{B}$ can induce special precovering classes in $(T\downarrow\mathscr{A})$. As an application, we prove that under suitable cases, the class of Gorenstein projective left $\Lambda$-modules over a triangular matrix ring $\Lambda=\left(\begin{smallmatrix}R & M \\ O & S \end{smallmatrix} \right)$ is special precovering if and only if both the classes of Gorenstein projective left $R$-modules and left $S$-modules are special precovering. Consequently, we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them., Comment: To appear in SCIENCE CHINA Mathematics

Orlov's famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. In this paper we show that this result is false without the full faithfulness hypothesis., Comment: 55 pages Appendix by Amnon Neeman

We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of $1$-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: we show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.

The paper studies analytic functors between presheaf categories. Generalising results of A. Joyal [11] and R. Hasegawa [9] for analytic endofunctors on the category of sets, we give two characterisations of analytic functors between presheaf categories over groupoids: (i) as functors preserving filtered colimits, quasi-pullbacks, and cofiltered limits; and (ii) as functors preserving filtered colimits and wide quasi-pullbacks. The development establishes that small groupoids, analytic functors between their presheaf categories, and quasi-cartesian natural transformations between them form a 2-category.

We generalize the definition of an exact sequence of tensor categories due to Brugui\`eres and Natale, and introduce a new notion of an exact sequence of (finite) tensor categories with respect to a module category. We give three definitions of this notion and show their equivalence. In particular, the Deligne tensor product of tensor categories gives rise to an exact sequence in our sense. We also show that the dual to an exact sequence in our sense is again an exact sequence. This generalizes the corresponding statement for exact sequences of Hopf algebras. Finally, we show that the middle term of an exact sequence is semisimple if so are the other two terms., Comment: 21 pages

A theorem by Orlov states that any equivalence \(F:D^{b}_{\mathrm {Coh}}(X) \rightarrow D^{b}_{\mathrm {Coh}}(Y)\) between the bounded derived categories of coherent sheaves of two smooth projective varieties X and Y is isomorphic to a Fourier–Mukai transform \(\Phi _{E}(-)=R\pi _{2*}(E\mathop {\otimes }\limits ^{L} L\pi _1^{*}(-))\), where the kernel E is in \(D^{b}_{\mathrm {Coh}}(X\times Y)\). In the case of an exact functor which is not necessarily fully faithful, we compute some sheaves that play the role of the cohomology sheaves of the kernel, and that are isomorphic to the latter whenever an isomorphism \(F\cong \Phi _{E}\) exists. We then exhibit a class of functors that are not full or faithful and still satisfy the above result.

We study the functorial properties of Bridgeland’s Hall algebras. Specifically, let 𝒜 and ℬ be two categories satisfying certain conditions for the definitions of Bridgeland’s Hall algebras, and let F:𝒜→ℬ be a fully faithful exact functor, which preserves projectives, then F induces an embedding of algebras from the Bridgeland’s Hall algebra of 𝒜 to the one of ℬ. In addition, let A be a finite-dimensional algebra over a finite field and B some special quotient algebra of A, then the Bridgeland’s Hall algebra of B is the quotient algebra of the one of A. Moreover, we consider the BGP-reflection functors on the category of 2-cyclic complexes and obtain some homomorphisms of algebras among the subalgebras of Bridgeland’s Hall algebras.

Let $G$ be a $\mathbb{Q}_p$-split reductive group with connected centre and Borel subgroup $B=TN$. We construct a right exact functor $D^\vee_\Delta$ from the category of smooth modulo $p^n$ representations of $B$ to the category of projective limits of finitely generated \'etale $(\varphi,\Gamma)$-modules over a multivariable (indexed by the set of simple roots) commutative Laurent-series ring. These correspond to representations of a direct power of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ via an equivalence of categories. Parabolic induction from a subgroup $P=L_PN_P$ corresponds to a basechange from a Laurent-series ring in those variables with corresponding simple roots contained in the Levi component $L_P$. $D^\vee_\Delta$ is exact and yields finitely generated objects on the category $SP_A$ of finite length representations with subquotients of principal series as Jordan-H\"older factors. Lifting the functor $D^\vee_\Delta$ to all (noncommuting) variables indexed by the positive roots allows us to construct a $G$-equivariant sheaf $\mathfrak{Y}_{\pi,\Delta}$ on $G/B$ and a $G$-equivariant continuous map from the Pontryagin dual $\pi^\vee$ of a smooth representation $\pi$ of $G$ to the global sections $\mathfrak{Y}_{\pi,\Delta}(G/B)$. We deduce that $D^\vee_\Delta$ is fully faithful on the full subcategory of $SP_A$ with Jordan-H\"older factors isomorphic to irreducible principal series., Comment: 55 pages, revised, to appear in Selecta Mathematica

The initial algebra for a set functor can be constructed iteratively via a well-known transfinite chain, which converges after a regular infinite cardinal number of steps or at most three steps. We extend this result to the analogous construction of relatively initial algebras. For the dual construction of the terminal coalgebra Worrell proved that if a set functor is α-accessible, then convergence takes at most α + α steps. But until now an example demonstrating that fewer steps may be insufficient was missing. We prove that the functor of all α-small filters is such an example. We further prove that for β ≤ α the functor of all α-small β-generated filters requires precisely α + β steps and that a certain modified power-set functor requires precisely α steps. We also present an example showing that whether a terminal coalgebra exists at all does not depend solely on the object mapping of the given set functor. (This contrasts with the fact that existence of an initial algebra is equivalent to existence of a mere fixed point.)

We find a new representation of the simple Lie algebra of type E 6 on the polynomial algebra in 16 variables, which gives a fractional representation of the corresponding Lie group on 16-dimensional space. Using this representation and Shen’s idea of mixed product, we construct a new functor from D 5-Mod to E 6-Mod. A condition for the functor to map a finite-dimensional irreducible D 5-module to an infinite-dimensional irreducible E 6-module is obtained. Our results yield explicit constructions of certain infinite-dimensional irreducible weight E6-modules with finite-dimensional weight subspaces. In our approach, the idea of Kostant’s characteristic identities plays a key role.