Applied Categorical Structures最新文献

(Completely regular) locales generalize (Tychonoff) spaces; indeed, the passage from a locale to its spatial sublocale is a well understood coreflection. But a locale also possesses an equally important pointless sublocale, and with morphisms suitably restricted, the passage from a locale to its pointless sublocale is also a coreflection. Our main theorem is that every locale can be uniquely represented as a subdirect product of its pointless and spatial parts, again with suitably restricted projections. We then exploit this representation by showing that any locale is determined by (what may be described as) the placement of its points in its pointless part.

We prove that the 2-category of action Lie groupoids localised in the following three different ways yield equivalent bicategories: localising at equivariant weak equivalences à la Pronk, localising using surjective submersive equivariant weak equivalences and anafunctors à la Roberts, and localising at all weak equivalences. These constructions generalise the known case of representable orbifold groupoids. We also show that any weak equivalence between action Lie groupoids is isomorphic to the composition of two particularly nice forms of equivariant weak equivalences.

It is well-known that DG-enhancements of the unbounded derived category ({text {D}}_{qc}(X)) of quasi-coherent sheaves on a scheme X are all equivalent to each other. Here we present an explicit model which leads to applications in deformation theory. In particular, we shall describe three models for derived endomorphisms of a quasi-coherent sheaf (mathcal {F}) on a finite-dimensional Noetherian separated scheme (even if (mathcal {F}) does not admit a locally free resolution). Moreover, these complexes are endowed with DG-Lie algebra structures, which we prove to control infinitesimal deformations of (mathcal {F}).

We categorify cocompleteness results of monad theory, in the context of pseudomonads. We first prove a general result establishing that, in any 2-category, weighted bicolimits can be constructed from oplax bicolimits and bicoequalizers of codescent objects. After prerequisites on pseudomonads and their pseudo-algebras, we give a 2-dimensional Linton theorem reducing bicocompleteness of 2-categories of pseudo-algebras to existence of bicoequalizers of codescent objects. Finally we prove this condition to be fulfilled in the case of a bifinitary pseudomonad, ensuring bicocompleteness.

We study Morita equivalence and Morita duality for rings with local units. We extend Auslander’s results on the theory of Morita equivalence and the Azumaya–Morita duality theorem to rings with local units. As a consequence, we give a version of Morita theorem and Azumaya–Morita duality theorem over rings with local units in terms of their full subcategory of finitely generated projective unitary modules and full subcategory of finitely generated injective unitary modules.

In this article, we prove Grothendieck’s Vanishing and Non-vanishing Theorems of local cohomology objects in the non-commutative algebraic geometry framework of Artin and Zhang. Let k be a field of characteristic zero and ({mathscr {S}}_{k}) be a strongly locally noetherian k-linear Grothendieck category. For a commutative noetherian k-algebra R, let ({mathscr {S}}_R) denote the category of R-objects in ({mathscr {S}}_k) obtained through a non-commutative base change by R of the abelian category ({mathscr {S}}_{k}). First, we establish Grothendieck’s Vanishing Theorem for any object ({mathscr {M}}) in ({mathscr {S}}_{R}). Further, if R is local and ({mathscr {S}}_{k}) is Hom-finite, we prove Non-vanishing Theorem for any finitely generated flat object ({mathscr {M}}) in ({mathscr {S}}_R).

Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system—which involves a continuous self-map on a metric space. There are many notions of complexity one can assign to the repeated iterations of the map. One of the foundational discoveries of dynamical systems theory is that these have a common limit, known as the topological entropy of the system. We present a category-theoretic view of topological dynamical entropy, which reveals that the common limit is a consequence of the structural assumptions on these notions. One of the key tools developed is that of a qualifying pair of functors, which ensure a limit preserving property in a manner similar to the sandwiching theorem from Real Analysis. It is shown that the diameter and Lebesgue number of open covers of a compact space, form a qualifying pair of functors. The various notions of complexity are expressed as functors, and natural transformations between these functors lead to their joint convergence to the common limit.

Suppose ((mathcal {C},mathbb {E},mathfrak {s})) is an n-exangulated category. We show that the idempotent completion and the weak idempotent completion of (mathcal {C}) are again n-exangulated categories. Furthermore, we also show that the canonical inclusion functor of (mathcal {C}) into its (resp. weak) idempotent completion is n-exangulated and 2-universal among n-exangulated functors from ((mathcal {C},mathbb {E},mathfrak {s})) to (resp. weakly) idempotent complete n-exangulated categories. Furthermore, we prove that if ((mathcal {C},mathbb {E},mathfrak {s})) is n-exact, then so too is its (resp. weak) idempotent completion. We note that our methods of proof differ substantially from the extriangulated and ((n+2))-angulated cases. However, our constructions recover the known structures in the established cases up to n-exangulated isomorphism of n-exangulated categories.

We will show that the Morrison–Walker blob complex appearing in Topological Quantum Field Theory is an operadic bar resolution of a certain operad composed of fields and local relations. As a by-product we develop the theory of unary operadic categories and study some novel and interesting phenomena arising in this context.

Right triangulated categories can be thought of as triangulated categories whose shift functor is not an equivalence. We give intrinsic characterisations of when such categories are appearing as the (co-)aisle of a (co-)t-structure in an associated triangulated category. Along the way, we also give an interpretation of these structures in the language of extriangulated categories.