We study the topological K-theory spectrum of the dg singularity category associated to a weighted projective complete intersection. We calculate the topological K-theory of the dg singularity category of a weighted projective hypersurface in terms of its Milnor fiber and monodromy operator, and, as an application, we obtain a lift of the Atiyah-Bott-Shapiro construction to the level of spectra.

This article is concerned with Alexandroff L-topological spaces and L-co-topological spaces, where L is a commutative, unital quantale. On one hand, an example is given to show that there is a finite strong L-topological space that is not Alexandroff. On the other hand, it is proved that every finite strong L-co-topological space is Alexandroff and that the category of Alexandroff strong L-co-t...

We give a complete and careful proof of Quillen’s theorem on the existence of the standard model category structure on the category of topological spaces. We do not assume any familiarity with model categories.

We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a fl...

For a symmetric monoidal-closed category X and any object K, the category of K-Chu spaces is small-topological over X and small cotopological over X . Its full subcategory of M-extensive K-Chu spaces is topological over X when X is Mcomplete, for any morphism class M. Often this subcategory may be presented as a full coreflective subcategory of Diers’ category of affine K-spaces. Hence, in addi...

In 1965, Zadeh [30] introduced a concept of a fuzzy set as the generalization of a crisp set. Also, in 1971, he introduced a fuzzy relation naturally, as a generalization of a crisp relation in [31]. Nel [27] introduced the notion of a topological universe which implies concrete quasitopos [1]. Every topological universe satisfies all the properties of a topos except one condition on the subobj...

Categories of lax (T, V )-algebras are shown to have pullbackstable coproducts if T preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the category of uniform spaces, are not presentable as a category of lax (T, V )-algebras, with T preserving inverse images. Moreov...

In the first part of this note an elementary proof is given of the fact that algebraic functors, that is, functors induced by morphisms of Lawvere theories, have left adjoints provided that the category K in which the models of these theories take their values is locally presentable. The main focus however lies on the special cases of the underlying functor of the category Grp(K) of internal gr...

Following the point-set lattice-theoretic (poslat) approach to topology of S. E. Rodabaugh [11], we introduced a more general way to do categorical (fuzzy) topology under the name of categorically-algebraic (catalg) topology [12], which unified many lattice-valued topological settings, and provided the means of intercommunication between different such frameworks. Moreover, motivated by the res...

The category TOP of topological spaces is not cartesian closed, but can be embedded into the cartesian closed category CONV of convergence spaces. It is well known that the category DCPO of dcpos and Scott continuous functions can be embedded into TOP, and so into CONV, by considering the Scott topology. We propose a di3erent, “cotopological” embedding of DCPO into CONV, which, in contrast to t...