Applied Categorical Structures最新文献

We show that the unbounded derived category of a Grothendieck category with enough projective objects is the base category of a derivator whose category of diagrams is the full 2-category of small categories. With this structure, we give a description of the localization functor associated to a specialization closed subset of the spectrum of a commutative noetherian ring. In addition, using the derivator of modules, we prove some basic theorems of group cohomology for complexes of representations over an arbitrary base ring.

We prove and explain several classical formulae for homotopy (co)limits in general (combinatorial) model categories which are not necessarily simplicially enriched. Importantly, we prove versions of the Bousfield–Kan formula and the fat totalization formula in this complete generality. We finish with a proof that homotopy-final functors preserve homotopy limits, again in complete generality.

({3/2})-Institutions have been introduced as an extension of institution theory that accommodates implicitly partiality of the signature morphisms together with its syntactic and semantic effects. In this paper we show that ordinary institutions that are equipped with an inclusion system for their categories of signatures generate naturally ({3/2})-institutions with explicit partiality for their signature morphisms. This provides a general uniform way to build ({3/2})-institutions for the foundations of conceptual blending and software evolution. Moreover our general construction allows for an uniform derivation of some useful technical properties.

We prove that for any small category ({mathcal {C}}), the category (textbf{KHausLoc}_{hat{{mathcal {C}}}}) of compact Hausdorff locales in the presheaf topos (hat{{mathcal {C}}}), is equivalent to the category of functors ({mathcal {C}} rightarrow textbf{KHausLoc}).

We introduce continuous analogues of Nakayama algebras. In particular, we introduce the notion of (pre-)Kupisch functions, which play a role as Kupisch series of Nakayama algebras, and view a continuous Nakayama representation as a special type of representation of ({mathbb {R}}) or ({mathbb {S}}^1). We investigate equivalences and connectedness of the categories of Nakayama representations. Specifically, we prove that orientation-preserving homeomorphisms on ({mathbb {R}}) and on ({mathbb {S}}^1) induce equivalences between these categories. Connectedness is characterized by a special type of points called separation points determined by (pre-)Kupisch functions. We also construct an exact embedding from the category of finite-dimensional representations for any finite-dimensional Nakayama algebra, to a category of continuous Nakayama representaitons.

A Pervin space is a set equipped with a bounded sublattice of its powerset, while its pointfree version, called Frith frame, consists of a frame equipped with a generating bounded sublattice. It is known that the dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames, and that the latter may be seen as representatives of certain quasi-uniform structures. As such, they have an underlying bitopological structure and inherit a natural notion of completion. In this paper we start by exploring the bitopological nature of Pervin spaces and of Frith frames, proving some categorical equivalences involving zero-dimensional structures. We then provide a conceptual proof of a duality between the categories of (T_0) complete Pervin spaces and of complete Frith frames. This enables us to interpret several Stone-type dualities as a restriction of the dual adjunction between Pervin spaces and Frith frames along full subcategory embeddings. Finally, we provide analogues of Banaschewski and Pultr’s characterizations of sober and (T_D) topological spaces in the setting of Pervin spaces and of Frith frames, highlighting the parallelism between the two notions.

Originally introduced in the context of the algebraic approach to term graph rewriting, the notion of gs-monoidal category has surfaced a few times under different monikers in the last decades. They can be thought of as symmetric monoidal categories whose arrows are generalised relations, with enough structure to talk about domains and partial functions, but less structure than cartesian bicategories. The aim of this paper is threefold. The first goal is to extend the original definition of gs-monoidality by enriching it with a preorder on arrows, giving rise to what we call oplax cartesian categories. Second, we show that (preorder-enriched) gs-monoidal categories naturally arise both as Kleisli categories and as span categories, and the relation between the resulting formalisms is explored. Finally, we present two theorems concerning Yoneda embeddings on the one hand and functorial completeness on the other, the latter inducing a completeness result also for lax functors from oplax cartesian categories to (textbf{Rel}).

We show that the quasicategory defined as the localization of the category of (simple) graphs at the class of A-homotopy equivalences does not admit colimits. In particular, we settle in the negative the question of whether the A-homotopy equivalences in the category of graphs are part of a model structure.

The theory of presentations of enriched monads was developed by Kelly, Power, and Lack, following classic work of Lawvere, and has been generalized to apply to subcategories of arities in recent work of Bourke–Garner and the authors. We argue that, while theoretically elegant and structurally fundamental, such presentations of enriched monads can be inconvenient to construct directly in practice, as they do not directly match the definitional procedures used in constructing many categories of enriched algebraic structures via operations and equations. Retaining the above approach to presentations as a key technical underpinning, we establish a flexible formalism for directly describing enriched algebraic structure borne by an object of a (mathscr {V})-category (mathscr {C}) in terms of parametrized (mathscr {J})-ary operations and diagrammatic equations for a suitable subcategory of arities (mathscr {J}hookrightarrow mathscr {C}). On this basis we introduce the notions of diagrammatic (mathscr {J})-presentation and (mathscr {J})-ary variety, and we show that the category of (mathscr {J})-ary varieties is dually equivalent to the category of (mathscr {J})-ary (mathscr {V})-monads. We establish several examples of diagrammatic (mathscr {J})-presentations and (mathscr {J})-ary varieties relevant in both mathematics and theoretical computer science, and we define the sum and tensor product of diagrammatic (mathscr {J})-presentations. We show that both (mathscr {J})-relative monads and (mathscr {J})-pretheories give rise to diagrammatic (mathscr {J})-presentations that directly describe their algebras. Using diagrammatic (mathscr {J})-presentations as a method of proof, we generalize the pretheories-monads adjunction of Bourke and Garner beyond the locally presentable setting. Lastly, we generalize Birkhoff’s Galois connection between classes of algebras and sets of equations to the above setting.

The category of internal groupoids (in an arbitrary category) is shown to be equivalent to the full subcategory of so called involutive-2-links that are unital and associative.