Abstract This paper concerns spherical adjunctions of stable $\infty $-categories and their relation to monadic adjunctions. We begin with a proof the 2/4 property in setting $-categories. The is based on description as $4$-periodic semiorthogonal decompositions given by Halpern-Leistner Shipman [4] Dyckerhoff et al. [3]. then describe class examples arising from local systems spheres. main res...

In [6, Theorem 2.2] Doi gave a Hopf-algebraic proof of a generalization of Oberst’s theorem on affine quotients of affine schemes. He considered a commutative Hopf algebra H over a field, coacting on a commutative H-comodule algebra A. If AcoH denotes the subalgebra of coinvariant elements of A and β : A ⊗AcoH A −→ A ⊗H the canonical map, he proved that the following are equivalent: (a) AcoH ⊂ ...

The theory of modules over associative algebras and the theory of comodules for coassociative coalgebras were developed fairly independently during the last decades. In this survey we display an intimate connection between these areas by the notion of categories subgenerated by an object. After a review of the relevant techniques in categories of left modules, applications to the bimodule struc...

Recognizable graph languages are a generalization of regular (word) languages to graphs (as well as arbitrary categories). Recently automaton functors were proposed as acceptors of recognizable graph languages. They promise to be a useful tool for the verification of dynamic systems, for example for invariant checking. Since automaton functors may contain an infinite number of finite state sets...

We introduce a new invariant for C?-algebras of stable rank one that merges the Cuntz semigroup information together with K1-group information. This semigroup, termed Cu1-semigroup, is constructed as equivalence classes pairs consisting positive element in stabilization given C?-algebra unitary unitization hereditary subalgebra generated by element. show Cu1-semigroup well-defined continuous fu...

This paper is the third paper of a series devoted to higherdimensional transition systems. The preceding paper proved the existence of a left determined model structure on the category of cubical transition systems. In this sequel, it is proved that there exists a model category of labelled symmetric precubical sets which is Quillen equivalent to the Bousfield localization of this left determin...

Let L/F be a finite separable field extension of degree n, X a smooth quasi-projective L-scheme, and R(X) the Weil transfer of X with respect to L/F . The map Z 7→ R(Z) of the set of simple cycles Z ⊂ X extends in a natural way to a map Z(X) → Z(R(X)) on the whole group of algebraic cycles Z(X). This map factors through the rational equivalence of cycles and induces this way a map of the Chow g...

A squarefree module over a polynomial ring S = k[x1, . . . , xn] is a generalization of a Stanley-Reisner ring, and allows us to apply homological methods to the study of monomial ideals more systematically. The category Sq of squarefree modules is equivalent to the category of finitely generated left Λ-modules, where Λ is the incidence algebra of the Boolean lattice 2. The derived category D(S...

For an ideal I of a preadditive category A, we study when the canonical functor C : A → A/I is local. We prove that there exists a largest full subcategory C of A for which the canonical functor C : C → C/I is local. Under this condition, the functor C turns out to be a weak equivalence between C and C/I. If A is additive (with splitting idempotents), then C is additive (with splitting idempote...