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# Homology, Homotopy and Applications

## Volume 8 (2006)

### Number 1

### Modelling fundamental 2-categories for directed homotopy

Pages: 31 – 70

DOI: https://dx.doi.org/10.4310/HHA.2006.v8.n1.a2

#### Author

#### 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.

#### Keywords

2-functors, local adjunctions, homotopy theory, directed algebraic topology, fundamental 2-category

#### 2010 Mathematics Subject Classification

18A40, 18D05, 55Pxx, 55Qxx, 55Uxx

Published 1 January 2006