Abstract In this study, mainly, we introduce the category Tex of textures and texturally continuous functions. Relations with some other categories, particulary dfTex, fTex, are pointed out. It is shown that the categories CTop of C-spaces and a full subcategory of Sober of sober spaces are, respectively, isomorphic to Tex and STex of simple textures. Some properties discussed including the exi...

We show that homotopy equivalent digital images have isomorphic fundamental groups, even when the homotopy equivalence does not preserve the basepoint. This assertion appeared in [3], but there was an error in the proof; here, we correct the error. We present and explore in detail a pair of digital images with cu-adjacencies that are homotopic but not pointed homotopic. For two digital loops f,...

We prove that the stable category associated with $$\mathsf {PreOrd}({\mathbb {C}})$$ of internal preorders in a pretopos $${\mathbb {C}}$$ satisfies universal property. The canonical functor from to {Stab}({\mathbb universally transforms pretorsion theory into classical torsion pointed . This also gives categorical insight construction first considered by Facchini and Finocchiaro special case ...

We describe two ways to define higher order Toda brackets in a pointed simplicial model category $${\mathcal {D}}$$ : one is recursive definition using categorical constructions, and the second uses associated enrichment. show that these definitions agree, by providing third, diagrammatic, description of bracket, explain how it serves as obstruction rectifying certain homotopy-commutative diagr...

We show that K1(E) of an exact category E agrees with K1(DE) of the associated triangulated derivator DE. More generally we show that K1(W) of a Waldhausen category W with cylinders and a saturated class of weak equivalences agrees with K1(DW) of the associated right pointed derivator DW. Introduction For a long time there was an interest in defining a nice K-theory for triangulated categories ...

We introduce a new notion of covering projection E → X of a topological spaceX which reduces to the usual notion ifX is locally connected. We use locally constant presheaves and covering reduced sieves to find a pro-groupoid π crs(X) and an induced category pro(π crs(X), Sets) such that for any topological space X the category of covering projections and transformations of X is equivalent to th...

We classify various types of graded extensions a finite braided tensor category $$\mathcal {B}$$ in terms its 2-categorical Picard groups. In particular, we prove that by group A correspond to monoidal 2-functors from the (consisting invertible central -module categories). Such functors can be expressed Eilnberg-Mac Lane cohomology. describe detail groups symmetric fusion categories and pointed...

Problem 50 in the Open Problems Project of the computational geometry community asks whether any triangulation on a point set in the plane contains a pointed spanning tree as a subgraph. We provide a counterexample. As a consequence we show that there exist triangulations which require a linear number of edge flips to become Hamiltonian.