We introduce a dihomotopy invariant of a po-space in dimension 2, its dihomotopy double category. This is a generalisation both of the fundamental category of a pospace, and of the double homotopy groupoid of a topological space. We conjecture a van Kampen theorem for the dihomotopy double category, thus making available a tool for calculations.

Since the early 1990's there have been several useful symmetric monoidal model structures on the underlying category of a point set category of spectra. Topological Hochschild homology (THH) is constructed from smash powers of ring-spectra, that is, monoids in the category of spectra. Already in the 1980's such a construction was made. Building on ideas of Goodwillie and Waldhausen, Bökstedt in...

In this note, we provide a quick introduction to the study of the Milnor fibration via the derived category and perverse sheaves. This is primarily a dictionary for translating from the standard topological setting to the derived category and/or the Abelian category of perverse sheaves.

For a stratified topological space we introduce the category of (mixed) IC-modules which are linear algebra devices with the relations described by the equation d = 0. We prove that the category of (mixed) IC-modules is equivalent to the category of (mixed) perverse sheaves for flag varieties.

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, ...

Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.

Let X be a connected scheme. Then one can associate (after Grothendieck) to X its algebraic fundamental group π1(X). This group π1(X) is a profinite group which is uniquely determined (up to inner automorphisms) by the property that the category of finite, discrete sets equipped with a continuous π1(X)-action is equivalent to the category of finite étale coverings of X. Moreover, the assignment...