By Marco Grandis

This can be the 1st authored publication to be devoted to the recent box of directed algebraic topology that arose within the Nineteen Nineties, in homotopy conception and within the conception of concurrent procedures. Its basic target might be acknowledged as 'modelling non-reversible phenomena' and its area could be individual from that of classical algebraic topology by way of the main that directed areas have privileged instructions and directed paths therein needn't be reversible. Its homotopical instruments (corresponding within the classical case to dull homotopies, primary team and basic groupoid) might be equally 'non-reversible': directed homotopies, primary monoid and basic classification. Homotopy buildings take place the following in a directed model, which supplies upward thrust to new 'shapes', like directed cones and directed spheres. functions will care for domain names the place privileged instructions seem, together with rewrite platforms, site visitors networks and organic platforms. the main constructed examples are available within the region of concurrency.

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**Additional info for Directed Algebraic Topology: Models of Non-Reversible Worlds (New Mathematical Monographs)**

**Example text**

Now, let A be concrete dI1-category; for the sake of simplicity, we assume that E = E op (which simply allows us to omit the speciﬁed isomorphism E → E op ). ∂ − = (∂ + )op : E → Iop . e. a map x : E → X, while a (directed) path in X is a map a : I → X, deﬁned on I = I(E); thus, a path is a homotopy on the standard point E, between its endpoints a : x− → x+ : E → X, xα = ∂ α (a) = a∂ α . 49) Every point x : E → A has a trivial path, 0x = xe : x → x (or degenerate path). ∂ +op = y op . (∂ α I) = xα e.

Ih. 43) (k ◦ ϕ ◦ h)op = k op ◦ ϕop ◦ hop : (kgh)op → (kf h)op (reﬂection). ) Actually, we can deﬁne a richer cubical structure on A (which will be important starting with Chapter 3): a p-dimensional homotopy ϕ : X →p Y is a map ϕˆ : I p X → Y . The composition with ψ : Y →q Z is n-dimensional (with n = p + q) and deﬁned as: ψ ◦ ϕ : X →n Z, ˆ q ϕˆ : I n X → I q Y → Z. 44) As we have already seen for topological spaces (cf. 11)), to extend the endofunctor I to homotopies requires a transposition s : I 2 → I 2 .

4 The basic structure of directed homotopies Let us examine the ﬁrst-order structure of the standard directed interval ↑I = ↑[0, 1], in the category pTop of preordered topological spaces. 19) where ↑I = {∗} is now an ordered space, with the unique order relation on the singleton. 20) as it takes values in the opposite preordered space ↑I (with the opposite preorder). 21) + c (t) = (t + 1)/2, since a mapping a : ↑I → X with values in a preordered space is a map (continuous and preorder-preserving) if and only if its two restrictions acα (to the ﬁrst or second half of the interval) are maps.