Left Exact Functors Research Articles

Auslander's defects over extriangulated categories: An application for the general heart construction

The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization of exact categories and triangulated categories. Our first aim is to provide an extension to extriangulated categories of Auslander's formula: for some extriangulated category $\mathcal{C}$, there exists a localization sequence $\operatorname{def}\mathcal{C} \to \mod\mathcal{C} \to \operatorname{lex}\mathcal{C}$, where $\operatorname{lex}\mathcal{C}$ denotes the full subcategory of finitely presented left exact functors and $\operatorname{def}\mathcal{C}$ the full subcategory of Auslander's defects. Moreover we provide a connection between the above localization sequence and the Gabriel–Quillen embedding theorem. As an application, we show that the general heart construction of a cotorsion pair $(\mathcal{U}, \mathcal{V})$ in a triangulated category, which was provided by Abe and Nakaoka, is the same as the construction of a localization sequence $\operatorname{def}\mathcal{U} \to \mod\mathcal{U} \to \operatorname{lex}\mathcal{U}$.

Eilenberg-Watts calculus for finite categories and a bimodule Radford 𝑆⁴ theorem

We obtain Morita invariant versions of Eilenberg-Watts type theorems, relating Deligne products of finite linear categories to categories of left exact as well as of right exact functors. This makes it possible to switch between different functor categories as well as Deligne products, which is often very convenient. For instance, we can show that applying the equivalence from left exact to right exact functors to the identity functor, regarded as a left exact functor, gives a Nakayama functor. The equivalences of categories we exhibit are compatible with the structure of module categories over finite tensor categories. This leads to a generalization of Radford’s S 4 S^4 -theorem to bimodule categories. We also explain the relation of our construction to relative Serre functors on module categories that are constructed via inner Hom functors.

Nakayama functors and Eilenberg-Watts theorems

In the present paper, analogs of the Eilenberg-Watts theorem are proved for categories of finitely generated modules over finite-dimensional algebras for right exact and left exact functors. Furthermore, for left exact functors the corresponding bimodules are described explicitly. The main aim of this paper is to present how, with these versions of the Eilenberg-Watts theorem, we obtain some new descriptions of the Nakayama functor and the inverse Nakayama functor in the case of self-injective algebras. Bibliography: 4 titles.

Derived Functors Related to Wall-Crossing

The setting is the representation theory of a simply connected, semisimple algebraic group over a field of positive characteristic. There is a natural transformation from the wall-crossing functor to the identity functor. The kernel of this transformation is a left exact functor. This functor and its first derived functor are evaluated on the global sections of a line bundle on the flag variety. It is conjectured that the derived functors of order greater than one annihilate the global sections. Also, the principal indecomposable modules for the Frobenius subgroups are shown to be acyclic.

On the derived DG functors

Assume that abelian categories A, B over a field admit countable direct limits and that these limits are exact. Let F : D dg(A)→ D + dg(B) be a DG quasi-functor such that the functor Ho(F) : D+(A)→ D+(B) carries D≥0(A) to D≥0(B) and such that, for every i > 0, the functor HiF : A → B is effaceable. We prove that F is canonically isomorphic to the right derived DG functor RH0(F). We also prove a similar result for bounded derived DG categories and a formula that expresses Hochschild cohomology of the categories Db dg(A), D + dg(A) as the Ext groups in the abelian category of left exact functors A → IndA . The proofs are based on a description of Drinfeld’s category of quasi-functors as the derived category of a category of sheaves.

A factorization of regularity

Earlier, the authors showed that categories with regular factorizations (in the sense of Kelly) and with regular epimorphisms closed under composition, are algebras for a KZ-doctrine R on the 2-category cat ker , of all categories and functors which preserve kernel inclusions. Here we show that the usual 2-category lex , of categories with finite limits and left exact functors, is cat ker L the 2-category of algebras for a coKZ-doctrine L on cat ker . Combined with our earlier results this shows that the 2-category reg , of regular categories and regular functors, is the 2-category of algebras for a distributive law LR → RL over the 2-category cat ker .

Some Spectral Sequences for Filtered Complexes and Applications

We construct a hierarchy of spectral sequences for a filtered complex under a left-exact functor. As applications we prove (1) the existence of a Leray spectral sequence for de Rham cohomology, (2) the equivalence of this sequence with the “usual” Leray spectral spectral sequence under the comparison isomorphism, and (3) the isomorphism of the Bloch–Ogus spectral sequence with the Leray spectral sequence for the morphism from the fine site to the Zariski site.

Constructive Complete Distributivity III

AbstractA complete lattice L is constructively completely distributive,(CCD)(L), if the sup map defined on down closed subobjects has a left adjoint. We characterize preservation of this property by left exact functors between toposes using a "logical comparison transformation". The characterization is applied to (direct images of) geometric morphisms to show that local homeomorphisms (in particular, product functors) preserve (CCD) objects, while preserving (CCD) objects implies openness.

Simply Connected Limits

The importance of finite limits in completeness conditions has been long recognized. One has only to consider elementary toposes, pretoposes, exact categories, etc., to realize their ubiquity. However, often pullbacks suffice and in a sense are more natural. For example it is pullbacks that are the essential ingredient in composition of spans, partial morphisms and relations. In fact the original definition of elementary topos was based on the notion of partial morphism classifier which involved only pullbacks (see [6]). Many constructions in topos theory, involving left exact functors, such as coalgebras on a cotriple and the gluing construction, also work for pullback preserving functors. And pullback preserving functors occur naturally in the subject, e.g. constant functors and the Σα. These observations led Rosebrugh and Wood to introduce partial geometric morphisms; functors with a pullback preserving left adjoint [9]. Other reasons led Kennison independently to introduce the same concept under the name semi-geometric functors [5].

Toposes and intuitionistic theories of types

It is shown that by relaxing the conventional notion of translation and introducing ‘comparisons’ of translations as 2-arrows, intuitionistic theories of types form a 2-category which is equivalent to the 2-category of toposes with left exact functors and natural transformations. In this equivalence, translations preserving disjunctions (resp. the existential quantifiers, logical translations) correspond to functors preserving unions (resp. images, logical functors).

On Representations of Grothendieck Toposes

Results of a representation-theoretic nature have played a major role in topos theory since the beginnings of the subject. For example, Deligne's theorem on coherent toposes, which says that every coherent topos has a continuous embedding into a topos of the form SetI for a discrete set I, is a typical result in the representation theory of toposes. (A continuous functor between toposes is the left adjoint of a geometric morphism. For Grothendieck toposes, it is exactly the same as a continuous functor between them, considered as sites with their canonical topologies. By a continuous functor between sites on left exact categories, we mean a left exact functor taking covers to covers.)A representation-like result for toposes typically asserts that a topos that satisfies some abstract conditions is related to a topos of some concrete kind; the relation between them is usually an embedding of the first topos in the second (concrete) one, for which the embedding satisfies some additional properties (fullness, etc.).

Some Results on Locally Finitely Presentable Categories

We prove that any full subcategory of a locally finitely presentable (l.f.p.) category having small limits and filtered colimits preserved by the inclusion functor is itself l.f.p. Here "full" may be weakened to "full with respect to isomorphisms." Further, we characterize those left exact functors $I:{\mathbf {C}} \to {\mathbf {D}}$ between small categories with finite limits for which the functor ${I^{\ast }}:{\mathbf {LEX}}({\mathbf {D}},{\text {Set)}} \to {\mathbf {LEX}}{\text {(}}{\mathbf {C}}{\text {,Set)}}$ induced by composition is full and faithful. As an application, we prove a theorem on sheaf representations, a consequence of which is that, for any site $\mathcal {C} = ({\mathbf {C}},J)$ on a category ${\mathbf {C}}$ with finite limits, defined by a subcanonical Grothendieck topology $J$, the closure in ${\mathbf {LEX}}({\mathbf {C}},{\text {Set)}}$ under small limits and filtered colimits of the models of $\mathcal {C}$ is the whole of ${\mathbf {LEX}}({\mathbf {C}},{\text {Set)}}$.

A Relationship between Left Exact and Representable Functors

Our aim in this paper is to demonstrate a relationship between left exact and representable functors. More precisely, in the functor category whose objects are the additive functors from the dual of an abelian category 𝔄 to the category of abelian groups and whose morphisms are the natural transformations between them, the left exact functors can be characterized as those equivalent to a direct limit of representable functors taken over a directed class. The proof will proceed in the following manner. Lambek [3] and Ulmer [7] have shown that any functor T in can be expressed as a direct limit of representable functors taken over a comma category. When T is left exact, it is easily observed that this comma category is a filtered category. When T is left exact, it is easily observed that this comma category is a filtered category.

On a cohomology theory for pairs of groups

of crossed homomorphisms from G to A vanishing on H. Clearly this is a left-exact functor in the category GMJF of left G-modules. The nth rightderived functor of X(G, H, -) in Ga9Z isdenoted byHn+'(G, H, -). The group Hn(G, H, A), A C GM, is called the nth cohomology group of the pair (G, H) with coefficients in A. These groups were first described and studied by M. Auslander in [1], who also found the sequence of Proposition 1.2. In this note we prove an excision property for the functors Hn(G, H, -), Theorem 2.2, and we find a direct sum decomposition of them under suitable conditions, Propositions 2.3 and 2.5. From this one deduces by standard methods a Mayer-Vietoris type sequence for the cohomology of groups, Proposition 2.6. The results in this paper are part of the author's doctoral dissertation at the University of Rochester. The author wishes to thank C. E. Watts for his advise and encouragement.