A notion of central importance in categorical topology is that of topological functor. A faithful functor E → B is called topological if it admits cartesian liftings of all (possibly large) families of arrows; the basic example is the forgetful functor Top→ Set. A topological functor E → 1 is the same thing as a (large) complete preorder, and the general topological functor E → B is intuitively...

This paper provides further developments in the study of the component categories which have been introduced in [7]. In particular, the component category functor is seen as a left adjoint hence preserves the pushouts. This property is applied to prove a Van Kampen like theorem for component categories. This last point is very important to make effective calculations. The original purpose of co...

In this paper we develop the theory of topological categories over a base category, that is, a theory of topological functors. Our notion of topo-logical functor is similar to (but not the same) the existing notions in the literature (see [2] 7.3), and it aims at the same examples. In our sense, a (pre) topological functor is a functor that creates cartesian families. A topological functor is, ...

Abstract We introduce the category of finite étale covers an arbitrary schematic space X and show that, equipped with appropriate natural fiber functor, it is a Galois Category. This allows us to define fundamental group spaces. If model scheme S , we that resulting theory on coincides classical therefore, recover introduced by Grothendieck. To prove these results, crucial find suitable geometr...

This article presents some theorems about functor structures. We start with some basic lemmata concerning the composition of functor structures. Then, two theorems about the restriction operator are formulated. Later, we show two theorems stating that the properties ’full’ and ’faithful’ of functor structures which are equivalent to the ’onto’ and ’one-to-one’ properties of their morphmaps, res...

We construct the quadratic analogue of the boson Fock functor. While in the first order (linear) case all contractions on the 1–particle space can be second quantized, the semigroup of contractions that admit a quadratic second quantization is much smaller due to the nonlinearity. The encouraging fact is that it contains, as proper subgroups (i.e. the contractions), all the gauge transformation...

In the paper [BB1], Beilinson and Bernstein used the method of localisation to give a new proof and generalisation of Casselman’s subrepresentation theorem. The key point is to interpret n-homology in geometric terms. The object of this note is to go one step further and describe the Jacquet module functor on Harish-Chandra modules via geometry. Let GR be a real reductive linear algebraic group...

Computability of Banach spaces is discussed. A compatible relation is shown to hold between the complex interpolation spaces of Calder) on (Studia Math. 24 (1964) 113) and the computability structures introduced by Pour-El and Richards (Computability in Analysis and Physics, Springer, Berlin, 1989). Namely, it is veri5ed that Calder) on’s original construction of the complex interpolation funct...

In this work, I address a primary issue with adapting categorical and algebraic concepts to functional analytic settings, the lack of free objects. Using a “normed set” and associated categories, I describe constructions of normed objects, which build from a set to a vector space to an algebra, and thus parallel the natural progression found in algebraic settings. Each of these is characterized...