By Marco Grandis
This can be the 1st authored ebook to be devoted to the hot box of directed algebraic topology that arose within the Nineties, in homotopy conception and within the conception of concurrent strategies. Its common objective could be said as 'modelling non-reversible phenomena' and its area can be special from that of classical algebraic topology via 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 boring homotopies, primary team and basic groupoid) might be equally 'non-reversible': directed homotopies, primary monoid and primary classification. Homotopy buildings happen right here in a directed model, which supplies upward thrust to new 'shapes', like directed cones and directed spheres. purposes will care for domain names the place privileged instructions seem, together with rewrite platforms, site visitors networks and organic structures. the main built examples are available within the region of concurrency.
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Additional resources for Directed Algebraic Topology: Models of Non-Reversible Worlds
Ip → idX), we say that X0 is a future (resp. past) deformation retract of X, that i is the embedding of a future (resp. past) deformation retract and that p is a future (resp. past) deformation retraction. 3 First-order homotopy theory by the cylinder functor, I 41 to each of these terms if one can choose the homotopy ϕ so that ϕi = 0i . e. it admits a retraction p with pi = id) and X0 is a regular subobject of X. 3). 58)), whence it also preserves future and past homotopy equivalences, as well as future and past deformation retracts.
63) In other words, R-duality amounts to applying the reversor R to objects and maps, and reﬂecting homotopies with the procedure ϕ → ϕop . 6), the dual of a small category is written as RX = X op . Small categories are viewed here as directed spaces, in their own right, and also as directed algebraic structures used to study other directed spaces via their fundamental categories. 8 Some forgetful functors and their adjoints The categories Top (of topological spaces) and pTop (of preordered topological spaces) have a canonical dIP1-structure, described in the previous section (the former is reversible).
The chaotic, or indiscrete, preorder). The standard embedding of Top in pTop will be the one given by the indiscrete preorder, so that all (ordinary) paths in S are directed in D S. Note that the category of ordered spaces does not allow for such an embedding, and would not allow us to view classical algebraic topology within the directed one; furthermore, its colimits are ‘diﬀerent’ from the topological ones. 2). Thus, every preordered space A having a locally compact Hausdorﬀ topology and an arbitrary preorder is exponentiable in pTop, with Y A consisting of the set pTop(A, Y ) ⊂ Top(U A, U Y ) of preorder-preserving continuous mappings, equipped with the (induced) compact-open topology and the pointwise preorder f ≺g if ( ∀ x ∈ A)(f (x) ≺Y g(x)).
Directed Algebraic Topology: Models of Non-Reversible Worlds by Marco Grandis