The Theory of Contexts is a type-theoretic axiomatization which has been recently proposed by some of the authors for giving a metalogical account of the fundamental notions of variable and context as they appear in Higher Order Abstract Syntax. In this paper, we prove that this theory is consistent by building a model based on functor categories. By means of a suitable notion of forcing, we pr...

In (non-commutative) geometry, a categorical resolution of a (potentially singular) variety X is a full and faithful embedding of its derived category D(X) into a smooth and proper triangulated category T [11, 12]. The notion generalises the situation of rational singularities, where the geometric resolution functor F : X̃ → X induces the full and faithful functor F ∗ : D(X) → D(X̃) to the smooth...

2 A bimodule realization of the Temperley-Lieb two-category 8 2.1 Ring A and two-dimensional cobordisms . . . . . . . . . . . . 8 2.2 Flat tangles and the Temperley-Lieb category . . . . . . . . . 10 2.3 The Temperley-Lieb 2-category . . . . . . . . . . . . . . . . . 12 2.4 The ring H . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Projective H-modules . . . . . . . . . . . . . . ....

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 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...

At the beginning of the 20th century there was an aim to generalize the non-commutative fundamental group to higher dimensions, hopes which seemed to be dashed in 1932 by the proof that the definition of higher homotopy groups πn then proposed by Čech led to commutative groups for n ≥ 2. Nonetheless, in the late 1930s and 1940s Whitehead developed properties of the second relative homotopy grou...

In terms of category theory, the Gromov homotopy principle for a set valued functor F asserts that the functor F can be induced from a homotopy functor. Similarly, we say that the bordism principle for an abelian group valued functor F holds if the functor F can be induced from a (co)homology functor. We examine the bordism principle in the case of functors given by bordism groups of solutions ...

Applicative functors [6] are a generalisation of monads. Both allow the expression of effectful computations into an otherwise pure language, like Haskell [5]. Applicative functors are to be preferred to monads when the structure of a computation is fixed a priori. That makes it possible to perform certain kinds of static analysis on applicative values. We define a notion of free applicative fu...

We study the circumstances under which one can reconstruct a stack from its associated functor of isomorphism classes. This is possible surprisingly often: we show that many of the standard examples of moduli stacks are determined by their functors. Our methods seem to exhibit new anabelian-type phenomena, in the form of structures in the category of schemes that encode automorphism data in gro...