By Marco Grandis
This is often the 1st authored ebook to be devoted to the recent box of directed algebraic topology that arose within the Nineteen Nineties, in homotopy thought and within the concept of concurrent strategies. Its common objective could be said as 'modelling non-reversible phenomena' and its area might be individual from that of classical algebraic topology by means of the primary 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 bland homotopies, basic team and basic groupoid) can be equally 'non-reversible': directed homotopies, basic monoid and primary class. Homotopy buildings ensue the following in a directed model, which provides upward push to new 'shapes', like directed cones and directed spheres. functions will take care of domain names the place privileged instructions look, together with rewrite structures, site visitors networks and organic platforms. the main constructed examples are available within the quarter of concurrency.
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Additional info for Directed Algebraic Topology: Models of Non-Reversible Worlds
Example text
11)), to extend the endofunctor I to homotopies requires a transposition s : I 2 → I 2 . 5. 4 Concrete structures A concrete dI1-category will be a dI1-category A equipped with a reversive object E, called the standard point, or free point, of A, and with a specified isomorphism E → E op . 7), invariant up to isomorphism under the reversor R U = | − | = A(E, −) : A → Set, UR ∼ = U. 46) 32 Directed structures and first-order homotopy properties Whenever possible, we will identify E = E op and U R = R.
1). 4, based on the reversor R(X) = X op (which reverses preorder) and the cylinder functor I(X) = X × ↑I. 6), R(X) = X op is the opposite category and the cylinder is I(X) = X × 2. 6). 4. 7 that they also admit a weakly reversible structure, where R is coherently isomorphic to the identity; the latter structure is also of interest, because it can be lifted to chain algebras (losing any reversibility property), while the first cannot. 5). Concatenation and the second-order structure will be dealt with in Part II.
A (directed) path a : 2 → X from x to x is an arrow a : x → x of X, their concatenation is the composition, strictly associative and unitary. This is a motivation of our frequent use of the additive notation for composition, as above for topological paths (see Section 8, in the Introduction). 2. 7). According to this adjunction, a (directed) homotopy 24 Directed structures and first-order homotopy properties ϕ : f → g : X → Y , represented by a functor X×2 → Y or, equivalently, X → Y 2 , is the same as a natural transformation f → g.