Applied Categorical Structures最新文献

In a previous paper we introduced the concept of semiseparable functor. Here we continue our study of these functors in connection with idempotent (Cauchy) completion. To this aim, we introduce and investigate the notions of (co)reflection and bireflection up to retracts. We show that the (co)comparison functor attached to an adjunction whose associated (co)monad is separable is a coreflection (reflection) up to retracts. This fact allows us to prove that a right (left) adjoint functor is semiseparable if and only if the associated (co)monad is separable and the (co)comparison functor is a bireflection up to retracts, extending a characterization pursued by X.-W. Chen in the separable case. Finally, we provide a semi-analogue of a result obtained by P. Balmer in the framework of pre-triangulated categories.

We study model structures on the category of comodules of a supercommutative Hopf algebra A over fields of characteristic 0. Given a graded Hopf algebra quotient (A rightarrow B) satisfying some finiteness conditions, the Frobenius tensor category ({mathcal {D}}) of graded B-comodules with its stable model structure induces a monoidal model structure on ({mathcal {C}}). We consider the corresponding homotopy quotient (gamma : {mathcal {C}} rightarrow Ho {mathcal {C}}) and the induced quotient ({mathcal {T}} rightarrow Ho {mathcal {T}}) for the tensor category ({mathcal {T}}) of finite dimensional A-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in (Ho {mathcal {T}}). We apply these results in the Rep(GL(m|n))-case and study its homotopy category (Ho {mathcal {T}}) associated to the parabolic subgroup of upper triangular block matrices. We construct cofibrant replacements and show that the quotient of (Ho{mathcal {T}}) by the negligible morphisms is again the representation category of a supergroup scheme.

We construct a category ({textrm{HomCob}}) whose objects are homotopically 1-finitely generated topological spaces, and whose morphisms are cofibrant cospans. Given a manifold submanifold pair (M, A), we prove that there exists functors into ({textrm{HomCob}}) from the full subgroupoid of the mapping class groupoid (textrm{MCG}_{M}^{A}), and from the full subgroupoid of the motion groupoid (textrm{Mot}_{M}^{A}), whose objects are homotopically 1-finitely generated. We also construct a family of functors ({textsf{Z}}_G:{textrm{HomCob}}rightarrow {textbf{Vect}}), one for each finite group G. These generalise topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf–Witten. Given a space X, we prove that ({textsf{Z}}_G(X)) can be expressed as the ({mathbb {C}})-vector space with basis natural transformation classes of maps from (pi (X,X_0)) to G for some finite representative set of points (X_0subset X), demonstrating that ({textsf{Z}}_G) is explicitly calculable.

This paper studies presentations of the Sierpinski gasket as a final coalgebra for a functor on three categories of metric spaces with additional designated points. The three categories which we study differ on their morphisms: one uses short (non-expanding) maps, the second uses continuous maps, and the third uses Lipschitz maps. The functor in all cases is very similar to what we find in the standard presentation of the gasket as an attractor. It was previously known that the Sierpinski gasket is bilipschitz equivalent (though not isomorhpic) to the final coalgebra of this functor in the category with short maps, and that final coalgebra is obtained by taking the completion of the initial algebra. In this paper, we prove that the Sierpiniski gasket itself is the final coalgebra in the category with continuous maps, though it does not occur as the completion of the initial algebra. In the Lipschitz setting, we show that the final coalgebra for this functor does not exist.

A locally coherent exact category is a finitely accessible additive category endowed with an exact structure in which the admissible short exact sequences are the directed colimits of admissible short exact sequences of finitely presentable objects. We show that any exact structure on a small idempotent-complete additive category extends uniquely to a locally coherent exact structure on the category of ind-objects; in particular, any finitely accessible category has the unique maximal and the unique minimal locally coherent exact category structures. All locally coherent exact categories are of Grothendieck type in the sense of Št’ovíček. We also discuss the canonical embedding of a small exact category into the abelian category of additive sheaves in connection with the locally coherent exact structure on the ind-objects, and deduce two periodicity theorems as applications.

Functorial semi-norms on singular homology measure the “size” of homology classes. A geometrically meaningful example is the (ell ^1)-semi-norm. However, the (ell ^1)-semi-norm is not universal in the sense that it does not vanish on as few classes as possible. We show that universal finite functorial semi-norms do exist on singular homology on the category of topological spaces that are homotopy equivalent to finite CW-complexes. Our arguments also apply to more general settings of functorial semi-norms.

We develop a hierarchy of semilattice bases (S-bases) for frames. For a given (unbounded) meet-semilattice A, we analyze the interval in the coframe of sublocales of the frame of downsets of A formed by all frames with the S-base A. We study various degrees of completeness of A, which generalize the concepts of extremally disconnected and basically disconnected frames. We introduce the concepts of D-bases and L-bases, as well as their bounded counterparts, and show how our results specialize and sharpen in these cases. Classic examples that are covered by our approach include zero-dimensional, completely regular, and coherent frames, allowing us to provide a new perspective on these well-studied classes of frames, as well as their spatial counterparts.

The Cohen–Macaulay Auslander algebra of an algebra A is defined as the endomorphism algebra of the direct sum of all indecomposable Gorenstein projective A-modules. The Cohen–Macaulay Auslander algebra of any string algebra is explicitly constructed in this paper. Moreover, it is shown that a class of special string algebras, which are called to be string algebras satisfying the G-condition, are representation-finite if and only if their Cohen–Macaulay Auslander algebras are representation-finite. As applications, it is proved that the derived representation type of gentle algebras coincide with their Cohen–Macaulay Auslander algebras.

We treat the problem of lifting bicategories into double categories through categories of vertical morphisms. We consider structures on decorated 2-categories allowing us to formally implement arguments of sliding certain squares along vertical subdivisions in double categories. We call these structures (pi _2)-indexings. We present a construction associating, to every (pi _2)-indexing on a decorated 2-category, a length 1 double internalization.

In this paper, we develop the notion of presentability in the parametrised homotopy theory framework of Barwick et al. (Parametrized higher category theory and higher algebra: a general introduction, 2016) over orbital categories. We formulate and prove a characterisation of parametrised presentable categories in terms of its associated straightening. From this we deduce a parametrised adjoint functor theorem from the unparametrised version, prove various localisation results, and we record the interactions of the notion of presentability here with multiplicative matters. Such a theory is of interest for example in equivariant homotopy theory, and we will apply it in Hilman (Parametrised noncommutative motives and cubical descent in equivariant algebraic K-theory, 2022) to construct the category of parametrised noncommutative motives for equivariant algebraic K-theory.