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We study torsion torsionfree(=TTF) triples in abelian and triangulated categories. (Notice that TTF triples in a triangulated category are essentially in bijection with recollement data for this triangulated category.) In particular, we…
This paper classifies spherical objects in various geometric settings in dimensions two and three, including both minimal and partial crepant resolutions of Kleinian singularities, as well as arbitrary flopping 3-fold contractions with only…
Relative theories(=closed subfunctors) are considered in exact, triangulated and extriangulated categories by Dr\"{a}xler-Reiten-Smal{\o}-Solberg-Keller, Beligiannis and Herschend-Liu-Nakaoka, respectively. We give a construction method of…
We define mutation pair in an n-angulated category and prove that given such a mutation pair, the corresponding quotient category carries a natural n-angulated structure. This result generalizes a theorem of Iyama-Yoshino in classical…
Let $\Cc$ and $\Dd$ be two corings over a ring $A$ and $\Cc\stackrel{\lambda}{\longrightarrow}\Dd$ be a morphism of corings. We investigate the situation when the associated induced ("corestriction of scalars") functor…
Maps (left adjoint arrows) between Frobenius objects in a cartesian bicategory B are precisely comonoid homomorphisms and, for A Frobenius and any T in B, map(B)(T,A) is a groupoid.
We introduce the dual notions of $\mathcal{E}(\mathcal{X},M,\mathcal{Y})$ and $\mathcal{M}(\mathcal{X},M,\mathcal{Y})$, and investigate when they have enough injective objects or projective objects, when they are resolving or co-resolving,…
Let D be a triangulated category with a cluster tilting subcategory U. The quotient category D/U is abelian; suppose that it has finite global dimension. We show that projection from D to D/U sends cluster tilting subcategories of D to…
We study abelian localizations of triangulated categories induced by rigid contravariantly finite subcategories, and also triangulated structures on subfactor categories of triangulated categories. In this context we generalize recent…
We define triangulated factorization systems on triangulated categories, and prove that a suitable subclass thereof (the normal triangulated torsion theories) corresponds bijectively to $t$-structures on the same category. This result is…
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…
A double category of relations is essentially a cartesian equipment with strong, discrete and functorial tabulators and for which certain local products satisfy a Frobenius Law. A double category of relations is equivalent to a double…
This paper provides geometrical descriptions of the Frobenius monad freely generated by a single object. These descriptions are related to results connecting Frobenius algebras and topological quantum field theories. In these descriptions,…
We introduce a spherical variant of Milnor's classifying construction for diffeological groups, based on quadratic normalization of barycentric coordinates. This construction gives rise to a contractible diffeological space endowed with…
Classification problems of subcategories have been deeply considered so far. In this paper, we discuss classifying dense (co)resolving subcategories of exact categories via their Grothendieck groups. This study is motivated by the…
Let C be a triangulated category with a Serre functor S and X a non-zero contravariantly finite rigid subcategory of C. Then X is cluster tilting if and only if the quotient category C/X is abelian and S(X)=X[2]. As an application, this…
The contributions of this paper are twofold. Within the framework of Grothendieck's fibrational category theory, we present a web of fundamental 2-adjunctions surrounding the formation of the category of all small diagrams in a given…
We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it preserves and reflects Gorenstein projective objects. We give conditions on when a Frobenius…
We study adjoint and Frobenius pairs of functors, equivalences, and the Picard group for corings.
A modular tensor category provides the appropriate data for the construction of a three-dimensional topological field theory. We describe the following analogue for two-dimensional conformal field theories: a 2-category whose objects are…