We describe an abstract framework in which the notion of fundamental category can be defined. The structures matching this framework are categories endowed with some additional structure. Provided we have a suitable adjunction between two of them, the fundamental categories defined in both cases can be easily compared. Each of these structures has a “natural” functor to the category of d-spaces...

We define a fundamental group for digital images. Namely, we construct functor from images to groups, which closely resembles the ordinary algebraic topology. Our construction differs in several basic ways previously established versions of setting. development gives prominent role subdivision show that our is preserved by subdivision.

We describe a fundamental additive functor Efund on the orbit category of a group. We prove that any isomorphism conjecture valid for Efund also holds for all additive functors, like K-theory, (topological) Hochschild or cyclic homology, etc. Finally, we reduce this universal isomorphism conjecture to K-theoretic ones, at the price of introducing some coefficients.

We study the category O for a general Coxeter system using a formulation of Fiebig. The translation functors, the Zuckerman functors and the twisting functors are defined. We prove the fundamental properties of these functors, the duality of Zuckerman functor and generalization of Verma’s result about homomorphisms between Verma modules.

Abstract We identify the exit path $\infty $-category of reductive Borel–Serre compactification as nerve a $1$-category defined purely in terms rational parabolic subgroups and their unipotent radicals. As immediate consequences, we fundamental group compactification, recovering result Ji–Murty–Saper–Scherk, obtain combinatorial incarnation constructible complexes sheaves on elements derived fu...

We study algebraic properties of categories of Merotopic, Nearness, and Filter Algebras. We show that the category of filter torsion free abelian groups is an epireflective subcategory of the category of filter abelian groups. The forgetful functor from the category of filter rings to filter monoids is essentially algebraic and the forgetful functor from the category of filter groups to the cat...

We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We ...

We study the representation theory of the invariant subalgebra of the Weyl algebra under a torus action, which we call a “hypertoric enveloping algebra.” We define an analogue of BGG category O for this algebra, and identify it with a certain category of sheaves on a hypertoric variety. We prove that a regular block of this category is highest weight and Koszul, identify its Koszul dual, comput...

In this paper, we give three functors $mathfrak{P}$, $[cdot]_K$ and $mathfrak{F}$ on the category of C$^ast$-algebras. The functor $mathfrak{P}$ assigns to each C$^ast$-algebra $mathcal{A}$ a pre-C$^ast$-algebra $mathfrak{P}(mathcal{A})$ with completion $[mathcal{A}]_K$. The functor $[cdot]_K$ assigns to each C$^ast$-algebra $mathcal{A}$ the Cauchy extension $[mathcal{A}]_K$ of $mathcal{A}$ by ...