Exact Category Research Articles

Personalized Image Classification by Semantic Embedding and Active Learning.

Currently, deep learning has shown state-of-the-art performance in image classification with pre-defined taxonomy. However, in a more real-world scenario, different users usually have different classification intents given an image collection. To satisfactorily personalize the requirement, we propose an interactive image classification system with an offline representation learning stage and an online classification stage. During the offline stage, we learn a deep model to extract the feature with higher flexibility and scalability for different users’ preferences. Instead of training the model only with the inter-class discrimination, we also encode the similarity between the semantic-embedding vectors of the category labels into the model. This makes the extracted feature adapt to multiple taxonomies with different granularities. During the online session, an annotation task iteratively alternates with a high-throughput verification task. When performing the verification task, the users are only required to indicate the incorrect prediction without giving the exact category label. For each iteration, our system chooses the images to be annotated or verified based on interactive efficiency optimization. To provide a high interactive rate, a unified active learning algorithm is used to search the optimal annotation and verification set by minimizing the expected time cost. After interactive annotation and verification, the new classified images are used to train a customized classifier online, which reflects the user-adaptive intent of categorization. The learned classifier is then used for subsequent annotation and verification tasks. Experimental results under several public image datasets show that our method outperforms existing methods.

Uniqueness of uniform decompositions in exact categories

Two uniqueness theorems on uniform decompositions due to Krause, Diracca and Facchini are extended from abelian categories to weakly idempotent complete exact categories. We give applications to (quasi-)abelian categories, finitely accessible additive categories and exactly definable additive categories.

On the Frobenius functor for symmetric tensor categories in positive characteristic

Abstract We develop a theory of Frobenius functors for symmetric tensor categories (STC) 𝒞 {\mathcal{C}} over a field 𝒌 {\boldsymbol{k}} of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F : 𝒞 → 𝒞 ⊠ Ver p {F:\mathcal{C}\to\mathcal{C}\boxtimes{\rm Ver}_{p}} , where Ver p {{\rm Ver}_{p}} is the Verlinde category (the semisimplification of Rep 𝐤 ( ℤ / p ) {\mathop{\mathrm{Rep}}\nolimits_{\mathbf{k}}(\mathbb{Z}/p)} ); a similar construction of the underlying additive functor appeared independently in [K. Coulembier, Tannakian categories in positive characteristic, preprint 2019]. This generalizes the usual Frobenius twist functor in modular representation theory and also the one defined in [V. Ostrik, On symmetric fusion categories in positive characteristic, Selecta Math. (N.S.) 26 2020, 3, Paper No. 36], where it is used to show that if 𝒞 {\mathcal{C}} is finite and semisimple, then it admits a fiber functor to Ver p {{\rm Ver}_{p}} . The main new feature is that when 𝒞 {\mathcal{C}} is not semisimple, F need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor 𝒞 → Ver p {\mathcal{C}\to{\rm Ver}_{p}} . We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of F, and use it to show that for categories with finitely many simple objects F does not increase the Frobenius–Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which F is exact, and define the canonical maximal Frobenius exact subcategory 𝒞 ex {\mathcal{C}_{\rm ex}} inside any STC 𝒞 {\mathcal{C}} with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius–Perron dimension is preserved by F. One of our main results is that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to Ver p {{\rm Ver}_{p}} . This is the strongest currently available characteristic p version of Deligne’s theorem (stating that a STC of moderate growth in characteristic zero is the representation category of a supergroup). We also show that a sufficiently large power of F lands in 𝒞 ex {\mathcal{C}_{\rm ex}} . Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category (namely, one having a fiber functor into the category of representations of the triangular Hopf algebra 𝒌 ⁢ [ d ] / d 2 {\boldsymbol{k}[d]/d^{2}} with d primitive and R-matrix R = 1 ⊗ 1 + d ⊗ d {R=1\otimes 1+d\otimes d} ), and show that a STC with Chevalley property is (almost) Frobenius exact. Finally, as a by-product, we resolve Question 2.15 of [P. Etingof and S. Gelaki, Exact sequences of tensor categories with respect to a module category, Adv. Math. 308 2017, 1187–1208].

Ideal cotorsion theories in triangulated categories

We study ideal cotorsion pairs associated to almost exact structures in extension closed subcategories of triangulated categories. This approach allows us to extend the recent ideal approximation theory developed by Fu, Herzog et al. for exact categories in the above mentioned context. In the last part of the paper we apply the theory in order to study projective classes (in particular localization or smashing subcategories) in compactly generated triangulated categories.

The acyclic closure of an exact category and its triangulation

For any exact category A with splitting idempotents, a maximal exact category T(A) containing A as a biresolving subcategory, is constructed. Important types of exact categories, including n-tilting torsion classes, categories of Cohen-Macaulay modules over a Cohen-Macaulay order, or categories of Gorenstein projectives, are shown to be of the form T(A). The quotient category T(A)/A in the sense of Grothendieck always exists and carries a triangulated structure. More generally, it is proved that any biresolving subcategory A of an exact category M gives a triangulated localization M/A. For example, the unbounded derived category D(A) of an exact category A is obtained directly, with no passage through a homotopy category of complexes. As applications, some recent developments related to Gorenstein projectivity, non-commutative crepant resolutions, singularity categories, and Cohen-Macaulay representations are extended and improved in the new framework. For example, the concept of non-commutative resolution of a noetherian Frobenius category is extended to arbitrary exact categories, which leads to an overarching connection with representation dimension of exact categories and n-tilting.

Different Exact Structures on the Monomorphism Categories

Let $${\mathcal {X}}$$ be a contravariantly finite resolving subcategory of $${\mathrm{{mod\text{- }}}}\varLambda $$ , the category of finitely generated right $$\varLambda $$ -modules. We associate to $${\mathcal {X}}$$ the subcategory $${\mathcal {S}}_{{\mathcal {X}}}(\varLambda )$$ of the morphism category $$\mathrm{H}(\varLambda )$$ consisting of all monomorphisms $$(A{\mathop {\rightarrow }\limits ^{f}}B)$$ with A, B and $$\text {Cok} f$$ in $${\mathcal {X}}$$ . Since $${\mathcal {S}}_{{\mathcal {X}}}(\varLambda )$$ is closed under extensions it inherits naturally an exact structure from $$\mathrm{H}(\varLambda )$$ . We will define two other different exact structures other than the canonical one on $${\mathcal {S}}_{{\mathcal {X}}}(\varLambda )$$ , and completely classify the indecomposable projective (resp. injective) objects in the corresponding exact categories. Enhancing $${\mathcal {S}}_{{\mathcal {X}}}(\varLambda )$$ with the new exact structure provides a framework to construct a triangle functor. Let $${\mathrm{{mod\text{- }}}}{\underline{{\mathcal {X}}}}$$ denote the category of finitely presented functors over the stable category $${\underline{{\mathcal {X}}}}$$ . We then use the triangle functor to show a triangle equivalence between the bounded derived category $${\mathbb {D}}^{\mathrm{b}}({\mathrm{{mod\text{- }}}}{\underline{{\mathcal {X}}}})$$ and a Verdier quotient of the bounded derived category of the associated exact category on $${\mathcal {S}}_{{\mathcal {X}}}(\varLambda )$$ . Similar consideration is also given for the singularity category of $${\mathrm{{mod\text{- }}}}{\underline{{\mathcal {X}}}}$$ .

A Hybrid CNN-Based Segmentation and Boosting Classifier for Real Time Sensor Spinal Cord Injury Data

Convolutional Neural network (CNN) based spinal cord disease prediction has emerged as a reliable model in medical imaging applications. Spinal cord injury (SCI) detection is one of the major problems for disorder segmentation and classification. Traditionally, radiologists analyze SCI images manually in order to detect abnormal spinal disorders. Manual interpretation of high dimensional feature space makes it difficult to predict the exact category and level of severity. On the other hand, deep learning framework helps to diagnose accurately and quickly. Deep learning approach is used to classify normal abnormal SCI images automatically. This paper presents a deep learning framework for helping diagnose SCI features based on the segmentation process. In this paper, a novel CNN-deep segmentation based boosting classifier is applied on sensor SCI image data. A real-time wearable sensor is used to capture the spinal cord disorder data with different shapes and orientations. Experimental results show that the present CNN-deep segmentation based boosting classifier has high computational SCI disorder prediction compared to the existing CNN based classifiers. Experimental results proved that the present model has better performance than the existing spinal cord injury detection models in terms of true positive rate; TP = 0.9859, Accuracy = 0.9894, and Error rate = 0.019 are concerned.

Phantom covering ideals in categories without enough projective morphisms

We give sufficient conditions to ensure that the ideal Φ(E′) of E′-phantom maps in a locally λ-presentable exact category (A,E) is a (special) (pre)covering ideal, where E′ is an exact substructure of (A,E). As a byproduct, we infer the existence of various covering ideals in categories of sheaves which have a meaningful geometrical motivation. In particular, we deal with a Zariski-local notion of phantom maps in categories of sheaves. We would like to point out that our approach is necessarily different from [18], as the categories involved in most of the examples we are interested in, do not have enough projective morphisms.

Dmathbb {Z}-Cluster tilting subcategories of singularity categories

For an exact category {{mathcal {E}}} with enough projectives and with a dmathbb {Z}-cluster tilting subcategory, we show that the singularity category of {{mathcal {E}}} admits a dmathbb {Z}-cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of {{mathcal {E}}} and {underline{{{mathcal {E}}}}}. We also deduce that the Gorenstein projectives of {{mathcal {E}}} admit a dmathbb {Z}-cluster tilting subcategory under some assumptions. Finally, we compute the dmathbb {Z}-cluster tilting subcategory of the singularity category for a finite-dimensional algebra which is not Iwanaga–Gorenstein.

Ziegler partial morphisms in additive exact categories

We develop a general theory of partial morphisms in additive exact categories which extends the model theoretic notion introduced by Ziegler in the particular case of pure-exact sequences in the category of modules over a ring. We relate partial morphisms with (co-)phantom morphisms and injective approximations and study the existence of such approximations in these exact categories.

Cotorsion pairs and a K-theory localization theorem

We show that a complete hereditary cotorsion pair (C,C⊥) in an exact category E, together with a subcategory Z⊆E containing C⊥, determines a Waldhausen category structure on the exact category C, in which Z is the class of acyclic objects.This allows us to prove a new version of Quillen's Localization Theorem, relating the K-theory of exact categories A⊆B to that of a cofiber. The novel idea in our approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed through a cotorsion pair. Notably, we do not require A to be a Serre subcategory, which produces new examples.Due to the algebraic nature of our Waldhausen categories, we are able to recover a version of Quillen's Resolution Theorem, now in a more homotopical setting that allows for weak equivalences.

The abelian closure of an exact category

The paper exhibits three constitutive exact subcategories of Adelman's free abelian category Ab(A) over a Quillen exact category A, which provide an intrinsic description of Ab(A) and encapsulate the mechanism of (finite or infinite) tilting. In general, two of these subcategories are left and right abelian, respectively, and a tilting adjunction takes place between them if they are abelian. The third category Im(A) contains the acyclic closureT(A) of A as an exact subcategory of those objects which arise as images of the morphisms in an exact complex over A. If the exact structure of A is trivial, T(A) consists of the Gorenstein projectives over A, a category that does not depend on an embedding of A into some ambient abelian category. The construction of Ab(A) and its intrinsic description sheds some light upon old and new concepts and results concerning Gruson-Jensen duality, resolving subcategories, representations of Cohen-Macaulay orders, Gorenstein projectivity, non-commutative resolutions, and representation dimension. Applications to infinite tilting and existence of derived equivalences will be given in a forthcoming publication.

RETRACTED ARTICLE: Hybrid method for mining rules based on enhanced Apriori algorithm with sequential minimal optimization in healthcare industry

Data mining may enable healthcare organizations, with analysis of the different prospects and connection between seemingly unrelated information, to anticipate trends in the patient’s medical condition and behavior. Raw data are large and heterogeneous from healthcare organizations. It needs to be collected and arranged, and its integration enables medical information systems to be integrated in a united way. Health data mining offers unlimited possibilities to evaluate numerous less obvious or secret data models utilizing common techniques for study. Association rule mining (ARM) is an effective technique for detecting the connection of the data which are the most commonly used and influential algorithms in ARM for an Apriori algorithm. However, it generates a large amount of rules and does not guarantee the efficiency and value of the knowledge created. In order to overcome this issue, an enhanced Apriori algorithm (EAA) based on the knowledge of a context ontology (EAA-SMO) methodology for sequential minimal optimization (SMO) is suggested. The simple knowledge is to establish the ideas of ontology as a hierarchical structure of the conceptual clusters of specific subjects, which comprises “similar” concepts that mean an exact category of the knowledge within the domain. There is an interesting rule for each cluster based on the correlation between the items. In addition, the rule developed is classified as a prediction model for anomaly detection based on SMO regression. The experimental analysis demonstrates the proposed method improved 2% of accuracy and minimizes the execution time by 25% when compared to semantic ontology.

Shortening binary complexes and commutativity of K-theory with infinite products

We show that in Grayson’s model of higher algebraic K-theory using binary acyclic complexes, the complexes of length two suffice to generate the whole group. Moreover, we prove that the comparison map from Nenashev’s model for K 1 K_1 to Grayson’s model for K 1 K_1 is an isomorphism. It follows that algebraic K K -theory of exact categories commutes with infinite products.

Relatively divisible and relatively flat objects in exact categories

We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair [Formula: see text] in an exact category [Formula: see text], [Formula: see text] coincides with the class of relatively flat objects of [Formula: see text] for some relative projectively generated exact structure, while [Formula: see text] coincides with the class of relatively divisible objects of [Formula: see text] for some relative injectively cogenerated exact structure. We exhibit Galois connections between relative cotorsion pairs in exact categories, relative projectively generated exact structures and relative injectively cogenerated exact structures in additive categories. We establish closure properties and characterizations in terms of the approximation theory.

Abelian quotients of categories of short exact sequences

We mainly investigate abelian quotients of categories of short exact sequences. The natural framework to consider the question is via identifying quotients of morphism categories as module categories. These ideas not only can be used to recover the abelian quotients produced by cluster-tilting subcategories of both exact categories and triangulated categories, but also can be used to reach our goal. Let (C,E) be an exact category. We denote by E(C) the category of bounded complexes whose objects are given by short exact sequences in E and by SE(C) the full subcategory formed by split short exact sequences. In general, E(C) is just an exact category, but the quotient E(C)/[SE(C)] turns out to be abelian. In particular, if (C,E) is Frobenius, we present three equivalent abelian quotients of E(C) and point out that the equivalences are actually given by left and right rotations. The abelian quotient E(C)/[SE(C)] admits some nice properties. We explicitly describe the abelian structure, projective objects, injective objects and simple objects, which provide a new viewpoint to understanding Hilton-Rees Theorem and Auslander-Reiten theory. Furthermore, we present some analogous results for triangulated versions.

Auslander’s defect formula and a commutative triangle in an exact category

We prove Auslander's defect formula in an exact category, and obtain a commutative triangle involving the Auslander bijections and the generalized Auslander-Reiten duality.

Proper classes and Gorensteinness in extriangulated categories

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. A notion of proper class in an extriangulated category is defined in this paper. Let C be an extriangulated category and ξ a proper class in C. We prove that C admits a new extriangulated structure. This construction gives extriangulated categories which are neither exact categories nor triangulated categories. Moreover, we introduce and study ξ-Gorenstein projective objects in C and demonstrate that ξ-Gorenstein projective objects share some basic properties with Gorenstein projective objects in module categories or in triangulated categories. In particular, we refine a result of Asadollahi and Salarian (2004) [1]. As an application, the admissible model structure on extriangulated categories is obtained.

Tilting subcategories in extriangulated categories

Extriangulated category was introduced by Nakaoka and Palu to give a unification of properties in exact categories and triangulated categories. A notion of tilting (or cotilting) subcategories in an extriangulated category is defined in this paper. We give a Bazzoni characterization of tilting (or cotilting) subcategories and obtain an Auslander-Reiten correspondence between tilting (cotilting) subcategories and coresolving covariantly (resolving contravariantly, resp.) finite subcatgories which are closed under direct summands and satisfies some cogenerating (generating, resp.) conditons. Applications of the results are given: we show that tilting (cotilting) subcategories defined here unify many previous works about tilting theory in module categories of Artin algebras and abelian categories admitting a cotorsion triples; we also show that the results work for triangulated categories with a proper class of triangles introduced by Beligiannis.

Abelian Categories Arising from Cluster Tilting Subcategories

For a triangulated category T, if C is a cluster-tilting subcategory of T, then the quotient category T\C is an abelian category. Under certain conditions, the converse also holds. This is an very important result of cluster-tilting theory, due to Koenig-Zhu and Beligiannis. Now let B be a suitable extriangulated category, which is a simultaneous generalization of triangulated categories and exact categories. We introduce the notion of pre-cluster tilting subcategory C of B, which is a generalization of cluster tilting subcategory. We show that C is cluster tilting if and only if B/C is abelian.