Applied Categorical Structures最新文献

The aim of this paper is to introduce the concept of n-Gorenstein tilting comodules and study its main properties. This concept generalizes the notion of n-tilting comodules of finite injective dimensions to the case of finite Gorenstein injective dimensions. As an application of our results, we discuss the problem of existence of complements to partial n-Gorenstein tilting comodules.

We show that certain pullbacks of (*)-algebras equivariant with respect to a compact group action remain pullbacks upon completing to (C^*)-algebras. This unifies a number of results in the literature on graph algebras, showing that pullbacks of Leavitt path algebras lift automatically to pullbacks of the corresponding graph (C^*)-algebras.

Commuting pairs of algebraic structures on a set have been studied by several authors and may be described equivalently as algebras for the tensor product of Lawvere theories, or more basically as certain bifunctors that here we call bifold algebras. The much less studied notion of commutant for Lawvere theories was first introduced by Wraith and generalizes the notion of centralizer clone in universal algebra. Working in the general setting of enriched algebraic theories for a system of arities, we study the interaction of the concepts of bifold algebra and commutant. We show that the notion of commutant arises via a universal construction in a two-sided fibration of bifold algebras over various theories. On this basis, we study special classes of bifold algebras that are related to commutants, introducing the notions of commutant bifold algebra and balanced bifold algebra. We establish several adjunctions and equivalences among these categories of bifold algebras and related categories of algebras over various theories, including commutative, contracommutative, saturated, and balanced algebras. We also survey and develop examples of commutant bifold algebras, including examples that employ Pontryagin duality and a theorem of Ehrenfeucht and Łoś on reflexive abelian groups. Along the way, we develop a functorial treatment of fundamental aspects of bifold algebras and commutants, including tensor products of theories and the equivalence of bifold algebras and commuting pairs of algebras. Because we work relative to a (possibly large) system of arities in a closed category ({mathscr {V}}), our main results are applicable to arbitrary ({mathscr {V}})-monads on a finitely complete ({mathscr {V}}), the enriched theories of Borceux and Day, the enriched Lawvere theories of Power relative to a regular cardinal, and other notions of algebraic theory.

(mathfrak {KNJ}) is the category of compact normal joinfit frames and frame homomorphisms and (mathfrak {KReg}) is the coreflective subcategory of compact regular frames. This work investigates (mathfrak {KNJ}) through its interaction with (mathfrak {KReg}) via the coreflection (rho ). A (mathfrak {KNJ}) morphism (phi : F longrightarrow M) is (mathcal {P})-essential if (phi ) is skeletal and the map between the frames of polars, (mathcal {P}(phi ): mathcal {P}F longrightarrow mathcal {P}M) defined by (mathcal {P}(phi )(p)=phi (p)^{perp perp }), is a boolean isomorphism. The (mathcal {P})-essential morphisms in (mathfrak {KNJ}) are closely related to the essential embeddings in (mathfrak {KReg}). We provide a characterization of the (mathcal {P})-essential morphisms in (mathfrak {KNJ}) and a connection to the essential embeddings in (mathfrak {KReg}). Further results about the preservation of joinfitness, the factorization of morphisms, and monomorphisms in (mathfrak {KNJ}) are provided. Moreover, in the category of (mathfrak {KNJ}) objects and skeletal frame homomorphisms, (mathfrak {KNJS}), we construct for (F in mathfrak {KNJ}) and (phi :rho F longrightarrow H) (an arbitrary (mathfrak {KReg}) essential embedding of (rho F)) the (mathfrak {KNJS}) pushout of (rho _F: rho F longrightarrow F) and (phi : rho F longrightarrow H). Lastly, we investigate the epimorphisms and epicomplete objects in (mathfrak {KNJS}).

In this paper, we introduce the notion of Grothendieck enriched categories for categories enriched over a sufficiently nice Grothendieck monoidal category (mathcal {V}), generalizing the classical notion of Grothendieck categories. Then we establish the Gabriel-Popescu type theorem for Grothendieck enriched categories. We also prove that the property of being Grothendieck enriched categories is preserved under the change of the base monoidal categories by a monoidal right adjoint functor. In particular, if we take as (mathcal {V}) the monoidal category of complexes of abelian groups, we obtain the notion of Grothendieck dg categories. As an application of the main results, we see that the dg category of complexes of quasi-coherent sheaves on a quasi-compact and quasi-separated scheme is an example of Grothendieck dg categories.

Let (Lambda ) be an artin algebra of finite global dimension. We study when the composition of three irreducible morphisms between indecomposable complexes in ({{mathbf {K}}^{b}(mathrm {proj},Lambda )}) is a non-zero morphism in the fourth power of the radical. We apply such results to prove that the composition of three irreducible morphisms between indecomposable complexes in the bounded derived category of a gentle Nakayama algebra, not selfinjective, whose ordinary quiver is an oriented cycle, belongs to the fourth power of the radical if and only if it vanishes.

We introduce the theory of enrichment over an internal monoidal category as a common generalization of both the standard theories of enriched and internal categories. Then, we contextualize the new notion by comparing it to another known generalization of enrichment: that of enrichment for indexed categories. It turns out that the two notions are closely related.

Let ({p : mathcal {E}rightarrow mathcal S}) be a hyperconnected geometric morphism. For each X in the ‘gros’ topos (mathcal {E}), there is a hyperconnected geometric morphism ({p_X : mathcal {E}/X rightarrow mathcal S(X)}) from the slice over X to the ‘petit’ topos of maps (over X) with discrete fibers. We show that if p is essential then (p_X) is essential for every X. The proof involves the idea of collapsing a connected subspace to a ‘basepoint’, as in Algebraic Topology, but formulated in topos-theoretic terms. In case p is local, we characterize when ({p_X}) is local for every X. This is a very restrictive property, typical of toposes of spaces of dimension ({le 1}).