Modelling fundamental 2-categories for directed homotopy

Abstract

Directed Algebraic Topology is a recent field, deeply linked with ordinary and higher dimensional Category Theory. A 'directed space', e.g. an ordered topological space, has directed homotopies (which are generally non-reversible) and fundamental n-categories (replacing the fundamental n-groupoids of the classical case). Finding a simple model of the latter is a nontrivial problem, whose solution gives relevant information on the given 'space'; a problem which is also of interest in general Category Theory, as it requires equivalence relations which are more general than categorical equivalence. Taking on a previous work on 'The shape of a category up to directed homotopy', we study now the fundamental 2-category of a directed space. All the notions of 2-category theory used here are explicitly reviewed.

The fundamental weighted category of a weighted space: From directed to weighted algebraic topology

“…This category is finitely complete and cocomplete, with the same finite limits and colimits as δMtr; but note that, now, the δ-metric of a (co)limit is only determined up to Lipschitz-equivalence. Moreover, δ ∞ Mtr is multiplicatively weighted [14], with the Lipschitz weight: all identities have ||1 X || 1 and composition gives ||gf || ||f ||.||g||. This also holds for δMtr, where ||f || 1.…”

“…All this forms the 2-category w + Cat of (small) w + -categories, also written wCat. (Here we do not use the multiplicative analogue, for which see [14]). …”

“…This is called a normed category in [18,2]; see the Introduction and [14] for the present terminology on the additive and multiplicative variants. We will omit the term 'additive' when there is no ambiguity; we also speak of a w + -category for short.…”

“…It forms a functor from the (obvious) category δMtr * of pointed δ-metric spaces, to the category of additively weighted monoids ( [14], 2.1)…”

“…Weighted categories can be viewed as categories enriched over the symmetric monoidal closed category w + Set of weighted sets (i.e., sets equipped with a mapping w : X → w + , and maps f : X → Y such that w(f (x)) w(x), for all x ∈ X), with an 'additive' tensor product [2,14].…”