Applied Categorical Structures最新文献

We give an elementary characterization of those quantaloids (mathcal {Q}) for which the category (textsf{Cat}(mathcal {Q})) of (mathcal {Q})-enriched categories and functors is cartesian closed. We then unify several known cases (previously proven using ad hoc methods) and we give some new examples.

We introduce derived projective covers and explain how they are related to the notion of enough derived projectives. This provides an if-and-only-if criterion for when derived projective covers form a silting collection. We prove moreover a Koszul duality result for silting and simple-minded collections.

We characterize projective objects in the category of internal crossed modules within any semi-abelian category. When this category forms a variety of algebras, the internal crossed modules again constitute a semi-abelian variety, ensuring the existence of free objects, and thus of enough projectives. We show that such a variety is not necessarily Schreier, but satisfies the so-called Condition (P)—meaning the class of projectives is closed under protosplit subobjects—if and only if the base variety satisfies this condition. As a consequence, the non-additive left chain-derived functors of the connected components functor are well defined in this context.

We provide an explicit construction of Hopf categories associated to comonoidal functors, generalizing Ševera’s construction of Hopf monoids through M-adapted functors. We discuss the example of the Hopf category whose underlying class is the set of twists of a Lie bialgebra. Finally, we apply the result to the setting of deformed categories.

In this paper, we introduce the cofibrant derived category of a group algebra kG and study its relation to the derived category of kG. We also define the cofibrant singularity category of kG, whose triviality characterizes the regularity of kG with respect to the cofibrant dimension, and examine its significance as a measure of the obstruction to the equality between the classes of Gorenstein projective and cofibrant modules. We show that the same obstruction can be measured by certain localization sequences between stable categories.

In this paper we introduce the notion of (pointed) prenormal category, modelled after regular categories, but with the key notions of coequaliser and kernel pair replaced by those of cokernel and kernel. This framework provides a natural setting for extending certain classical results in algebra. We study the fundamental properties of prenormal categories, including a characterisation in terms of a factorisation system involving normal epimorphisms, and a categorical version of Noether’s so-called ‘third isomorphism theorem’. We also present a range of examples, with the category of commutative monoids constituting a central one. In the second part of the paper we extend prenormality and its related properties to the non-pointed context, using kernels and cokernels defined relative to a distinguished class of trivial objects.

In this paper, we provide a comprehensive analysis of involutive quantales, with a particular focus on quantic frames. We extend the axiomatic foundations of quantale-enriched topological spaces to include closure under the anti-homomorphic involution, facilitating a balanced topologization of the spectrum of unital (C^*)-algebras that encompasses both closed right and left ideals through the concept of quantic frames. Specifically, certain subspaces of pure states are identified as strongly Hausdorff separated, involutive quantale-enriched topological spaces.

In this paper we present a unified proof of the fact that the category of modules over a ring and the category of near-vector spaces in the sense of J. André, over an appropriate scalar system (a ‘scalar group’), are both abelian categories. The unification is possible by viewing each of these categories as subcategories of the (abelian) category of modules over a multiplicative monoid M. Although in the case of near-vector spaces all elements of M except one (the ‘zero’ element) are invertible, we show that this requirement is not necessary for the corresponding category to be abelian in analogy to the well-known fact that modules over a ring form an abelian category even if the ring is not a field (i.e., modules over it are not vector spaces).

Given a functor (F: mathcal {C}rightarrow mathcal {D}) and a model-theoretic independence relation on (mathcal {D}), we can lift that independence relation along F to (mathcal {C}) by declaring a commuting square in (mathcal {C}) to be independent if its image under F is independent. For each property of interest that an independence relation can have we give assumptions on the functor that guarantee the property to be lifted.

We prove some facts about locales L equipped with the Scott topology ({Omega }(L)), in particular studying a canonical frame homomorphism (phi :{Omega }(L)rightarrow L) which is motivated by an application to cognitive science. Such a topological locale L is called a Scott locale if the inclusion of primes (p:{Sigma }(L)rightarrow L) is continuous. We prove that the spectrum ({Sigma }(L)) of a Scott locale L is necessarily (T_1), and that preregular locales (a generalization of regular locales) are Scott locales. If L is the topology of a topological space X we find a (necessarily unique) continuous map (f:Xrightarrow L) such that (f^{-1}=phi ) and compare it with the points-to-primes map (p:Xrightarrow L), showing that (f=p) if and only if X is preregular, and that a sober space X is Hausdorff if and only if X is (T_1) and (f(X)subseteq {Sigma }(L).)