By Marco Grandis

ISBN-10: 0521760364

ISBN-13: 9780521760362

This is often the 1st authored e-book to be devoted to the recent box of directed algebraic topology that arose within the Nineties, in homotopy thought and within the concept of concurrent methods. Its normal objective could be said as 'modelling non-reversible phenomena' and its area could be wonderful from that of classical algebraic topology via the main that directed areas have privileged instructions and directed paths therein don't need to be reversible. Its homotopical instruments (corresponding within the classical case to dull homotopies, basic team and basic groupoid) can be equally 'non-reversible': directed homotopies, primary monoid and primary class. Homotopy buildings ensue right here in a directed model, which supplies upward push to new 'shapes', like directed cones and directed spheres. purposes will care for domain names the place privileged instructions look, together with rewrite structures, site visitors networks and organic structures. the main constructed examples are available within the region of concurrency.

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**Additional resources for Directed Algebraic Topology: Models of Non-Reversible Worlds**

**Example text**

Most of this material has been introduced in [G1, G3, G4, G21]. The starting point, in the classical (reversible) case, goes back to Kan’s abstract cylinder approach [Ka2]. 23) for preordered spaces. e. ∂ − R = R∂ + : R → RI. 37) Since RR = 1, the transformation r is invertible with r−1 = RrR : RI → IR. ∂ + R = R∂ − . ∂ α X = f α . When we want to distinguish the homotopy from the map which represents it, we write the latter as ϕ. If : IX → Y . 38) and (ϕop )op = ϕ, (0f )op = 0f o p . An object X is said to be reversive or self-dual if it is isomorphic to op X .

We have already said that directed homotopy equivalence in Cat will be studied later. Ordinary equivalence of categories is a stricter, far simpler notion. 30) with the obvious reversion r, deﬁned above. This gives rise to a reversible cylinder functor X ×i, with right adjoint Y i (the full subcategory of Y 2 whose objects are the isomorphisms of Y ); thus, a reversible homotopy ϕ : f → g : X → Y is the same as a natural isomorphism of functors. This reversible homotopy structure will be written as Cati .

The presence of the transposition symmetry, for preordered spaces and d-spaces, reveals that the directed character of these structures does not go beyond the one-dimensional level: after distinguishing some paths ↑I → X and forbidding others, no higher choice is needed: namely, a n continuous mapping a : ↑I → X (in pTop or dTop) is a map of the n category if and only if, for every increasing map f : ↑I → ↑I , the path af : ↑I → X is a map. 1 From classical homotopy to the directed case 23 we can also notice that an element of Kn need not have any counterpart with faces permuted (for n 2).

### Directed Algebraic Topology: Models of Non-Reversible Worlds by Marco Grandis

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