# Inessential directed maps and directed homotopy equivalences

@article{Raussen2021InessentialDM,
title={Inessential directed maps and directed homotopy equivalences},
author={Martin Raussen},
journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics},
year={2021},
volume={151},
pages={1383 - 1406}
}
• M. Raussen
• Published 21 June 2019
• Mathematics
• Proceedings of the Royal Society of Edinburgh: Section A Mathematics
A directed space is a topological space $X$ together with a subspace $\vec {P}(X)\subset X^I$ of directed paths on $X$. A symmetry of a directed space should therefore respect both the topology of the underlying space and the topology of the associated spaces $\vec {P}(X)_-^+$ of directed paths between a source ($-$) and a target ($+$)—up to homotopy. If it is, moreover, homotopic to the identity map—in a directed sense—such a symmetry will be called an inessential d-map, and the paper explores…
1 Citations

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## References

SHOWING 1-10 OF 21 REFERENCES

### On directed homotopy equivalences and a notion of directed topological complexity

This short note introduces a notion of directed homotopy equivalence and of "directed" topological complexity (which elaborates on the notion that can be found in e.g. Farber's book) which have a

### Pair component categories for directed spaces

A pair component category as quotient category is constructed: it has as objects pair components along which the homotopy type is invariant—for a coherent and transparent reason and gives reasonable results for spaces with non-trivial directed loops.

### Directed homotopy theory, I. The fundamental category

Directed Algebraic Topology is beginning to emerge from various applications. The basic structure we shall use for such a theory, a 'd-space', is a topological space equipped with a family of

### Directed Algebraic Topology: Models of Non-Reversible Worlds

This is the first authored book to be dedicated to the new field of directed algebraic topology that arose in the 1990s, in homotopy theory and in the theory of concurrent processes. Its general aim

### Algebraic topology and concurrency

• Mathematics, Computer Science
Theor. Comput. Sci.
• 2006

### Topological Complexity of Motion Planning

• M. Farber
• Mathematics
Discret. Comput. Geom.
• 2003
The topological complexity of the problem of motion planning for a robot arm in the absence of obstacles is completely calculated.

### Invariants of Directed Spaces

• M. Raussen
• Mathematics
Appl. Categorical Struct.
• 2007
This work studies the preorder category associated to a directed space and various “quotient” categories arising from algebraic topological functors and proposes and studies a new notion of directed homotopy equivalence between directed spaces.

### Reparametrizations of continuous paths

• Mathematics
• 2007
A reparametrization (of a continuous path) is given by a surjective weakly increasing self-map of the unit interval. We show that the monoid of reparametrizations (with respect to compositions) can

### Stable Components of Directed Spaces

It is shown that the geometric realizations of finite pre-cubical sets with no loops admit unique minimal stable (future/past/total) component systems, which provide a new family of invariants for directed spaces.