Comparison of categories of traces in directed topology
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1 Comparison of categories of traces in directed topolog Abel Smposium, Geiranger 2018 Martin Raussen Department of Mathematical Sciences Aalborg Universit
2 Directed Algebraic Topolog Data: D-spaces Result: Dipaths and traces 1 Definition A d-space (M. Grandis) is a topological space X together with a subspace P(X ) (the dipaths) of path space X I that contains all constant paths is closed under concatenation is closed under non-decreasing reparametrizations I I
3 Directed Algebraic Topolog Data: D-spaces Result: Dipaths and traces 1 Definition A d-space (M. Grandis) is a topological space X together with a subspace P(X ) (the dipaths) of path space X I that contains all constant paths is closed under concatenation is closed under non-decreasing reparametrizations I I Glossar Trace = Path up to non-decreasing onto reparametrization P(X ) x : Dipaths from x to (CO-topolog) T (X ) x : Traces from x to (quotient topolog) OBS: The end point map q : T (X ) X X is not a fibration, in general!
4 Trace spaces Pre-cubical sets 2 Pre-cubical sets and their geometric realization: glued from cubes along subcubes (like for pre-simplicial sets). D-structure: Paths that are non-decreasing along ever coordinate. Glued together at common boundaries.
5 Trace spaces Pre-cubical sets 2 Pre-cubical sets and their geometric realization: glued from cubes along subcubes (like for pre-simplicial sets). D-structure: Paths that are non-decreasing along ever coordinate. Glued together at common boundaries. For pre-cubical sets, there exist polhedral spaces homotop equivalent to T (X ) x finite if X does not admit non-trivial directed loops (MR, K. Ziemiański). Example: X = n * T ( n ) is empt or contractible or homotop equivalent to S k, 0 k n 2. S 0 S 1
6 Primar motivation: Concurrenc A simple PV program: Swiss flag 3 P 2 1 Vb Va Pa Pb Unsafe Unreachable 0 Pa Pb Vb Va P 1
7 Primar motivation: Concurrenc A simple PV program: Swiss flag 3 P 2 Vb Va Pa Pb 0 Unsafe Unreachable Pa Pb Vb Va 1 P 1 Executions are directed paths since time flow is irreversible avoiding a forbidden region (shaded). Dipaths that are dihomotopic (through a 1-parameter deformation consisting of dipaths) correspond to equivalent executions.
8 2 categories associated with a d-space X 1. Trace categor 4 Trace categor T (X ) Objects Points x X Morphisms Traces T (X ) x for x, i.e., T (X ) x = Identities trivial dipath σ x T (X ) x x Composition Concatenation : T (X ) x T (X ) z T (X ) z x
9 2 categories associated with a d-space X 1. Trace categor 4 Trace categor T (X ) Objects Points x X Morphisms Traces T (X ) x for x, i.e., T (X ) x = Identities trivial dipath σ x T (X ) x x Composition Concatenation : T (X ) x T (X ) z T (X ) z x Analog: Poset categor from filtered space X = r R X r Objects r R Morphism r s Identities r = r Composition r s t
10 2 categories associated with a d-space X 1. Trace categor 4 Trace categor T (X ) Objects Points x X Morphisms Traces T (X ) x for x, i.e., T (X ) x = Identities trivial dipath σ x T (X ) x x Composition Concatenation : T (X ) x T (X ) z T (X ) z x Analog: Poset categor from filtered space X = r R X r Objects r R Morphism r s Identities r = r Composition r s t Appl algebraic topolog functors to HoTop, Ab etc. Which morphisms go to isos? Morphisms between certain pairs of objects ma be empt.
11 2 categories associated with a d-space X 2. Its envelopping categor (extension categor) 5 Extension categor E T (X ) Objects Pairs (x, ), x Morphisms (extension) maps (τ x x, τ ) T (X ) x T (X ) Identities (σ x, σ ) Composition Concatenation in future and past Subcategories Onl future (+), onl past (-). x
12 2 categories associated with a d-space X 2. Its envelopping categor (extension categor) 5 Extension categor E T (X ) Objects Pairs (x, ), x Morphisms (extension) maps (τ x x, τ ) T (X ) x T (X ) Identities (σ x, σ ) Composition Concatenation in future and past Subcategories Onl future (+), onl past (-). x Analog: Filtered space Objects Pairs (r, s), r s Morphism (r, s) (r, s ) r r, s s Identit Equalities (source & target) Composition of
13 2 categories associated with a d-space X 2. Its envelopping categor (extension categor) 5 Extension categor E T (X ) Objects Pairs (x, ), x Morphisms (extension) maps (τ x x, τ ) T (X ) x T (X ) Identities (σ x, σ ) Composition Concatenation in future and past Subcategories Onl future (+), onl past (-). Functor E T (X ) HoTop x Analog: Filtered space Objects Pairs (r, s), r s Morphism (r, s) (r, s ) r r, s s Identit Equalities (source & target) Composition of (x, ) T (X ) x (τ x x, τ ) [σ T (X ) x τ x x σ τ T (X ) x ]
14 Goal: Find small equivalent categories Two strategies 6 First explained for filtered space: 1. Replace reals b maximal (ordered disjoint) intervals J such that X r X s is a (weak) homotop equivalence (resp. induces an iso of algebraic invariant) for all r, s J, r s. Morphism: I J maps to homotop class of X r X s up to homotop equivalence. 2. Pick representatives in each of these intervals and consider the full subcategor on onl these objects.
15 Goal: Find small equivalent categories Two strategies 6 First explained for filtered space: 1. Replace reals b maximal (ordered disjoint) intervals J such that X r X s is a (weak) homotop equivalence (resp. induces an iso of algebraic invariant) for all r, s J, r s. Morphism: I J maps to homotop class of X r X s up to homotop equivalence. 2. Pick representatives in each of these intervals and consider the full subcategor on onl these objects. The reals categor and the resulting enriched poset categories are equivalent: 1. The resulting quotient map is full faithful (in the enriched sense) and surjective on objects. 2. The subcategor is full faithful and essentiall surjective on objects.
16 Example: 2D-Mutual exclusion Trace categor 7 B 2 + C + A - - B 1
17 Example: 2D-Mutual exclusion Trace categor 7 Compressed trace categor B 1 B 2 + C A C + B 2 A - - B 1 taking into account both composition in the past and in the future. Can compress to two objects if composition is considered onl in the future (resp. onl in the past).
18 Example: 2D-Mutual exclusion Extension categor 8 Compressed extension categor B 2 + C + B 1 B 1 + AB 1 B 1 C + + AA AC CC + + A - - B 1 AB 2 + B 2 B 2 B 2 C homolog functor = natural homolog (Dubut-Goubault ) Necessar?
19 Example: A cubical complex Glue along blue and red edges 9 Modification of an example from J. Dubut s thesis (2017): B 2 C A B 1
20 Example: A cubical complex Glue along blue and red edges 9 Modification of an example from J. Dubut s thesis (2017): B 2 C A B 1 Directed paths? Increasing in each cube. Fit at glueings.
21 Example: A cubical complex No dipath from to 10 C B 2 A B 1
22 Example: A cubical complex One dipath from to (through B 1 ) 11 B 2 C A B 1
23 A cubical complex One dipath from to (through B 2 ) 12 B 2 C A B 1
24 Example: A cubical complex Two dipaths from to (through B 1 resp. B 2 ) 13 B 2 C A B 1
25 Example: A cubical complex Two dipaths from to (through B 1 resp. B 2 ) 13 B 2 C A B 1 Lesson The relative position of and of (within their cubes) is decisive for the homotop tpe of T (X )!
26 Example: A cubical complex No isomorphisms in the trace categor 14 B 2 C A B 1 one dipath through B 1 one dipath through B 2 two dipaths
27 Example: A cubical complex No isomorphisms in the trace categor 14 B 2 C A B 1 one dipath through B 1 one dipath through B 2 two dipaths
28 Example: A cubical complex No isomorphisms in the trace categor 14 B 2 C A B 1 one dipath through B 1 one dipath through B 2 two dipaths
29 Example: A cubical complex No isomorphisms in the trace categor 14 B 2 A C B 1 OBS: No non-trivial dipath σ within the cell C preserves ever path space up to homotop equivalence: For ever such σ there exists a A such that σ : σ(0) P(X ) a σ(1) P(X ) a is not a homotop equivalence. one dipath through B 1 one dipath through B 2 two dipaths
30 Example: A cubical complex No isomorphisms in the trace categor 14 B 2 A C B 1 one dipath through B 1 one dipath through B 2 two dipaths OBS: No non-trivial dipath σ within the cell C preserves ever path space up to homotop equivalence: For ever such σ there exists a A such that σ : σ(0) P(X ) a σ(1) P(X ) a is not a homotop equivalence. Trace categor cannot be compressed!
31 Example: A cubical complex Better luck with the extension categor 15 Compressed extension categor B1B1 + AB1 AC+ B1C AC+ + + AA AC++ CC + + B 2 C AB2 AC + + B2B2 B2C AC + A B 1 AC + : 1. coord. decreases, 2. coord. increases AC + : 1. coord. increases, 2. coord. decreases AC ++ : Both coord. increase
32 Example: A cubical complex Better luck with the extension categor 15 Compressed extension categor B1B1 + AB1 AC+ B1C AC+ + + AA AC++ CC + + B 2 C AB2 AC + + B2B2 B2C AC + A B 1 AC + : 1. coord. decreases, 2. coord. increases AC + : 1. coord. increases, 2. coord. decreases AC ++ : Both coord. increase Isomorphic to the extension categor from previous example
33 Path space preserving homotop flows 16 Definition (Psp homotop flow) A homotop flow on a d-space X is a directed homotop H : X I X with H 0 = id X. A homotop flow is path space preserving (psp) if, for all x and all t I, the maps H(, t) induce homotop equivalences H t (x, ) : T (X ) x T (X ) H(,t) H(x,t), σ H t σ
34 Path space preserving homotop flows 16 Definition (Psp homotop flow) A homotop flow on a d-space X is a directed homotop H : X I X with H 0 = id X. A homotop flow is path space preserving (psp) if, for all x and all t I, the maps H(, t) induce homotop equivalences H t (x, ) : T (X ) x T (X ) H(,t) H(x,t), σ H t σ (Psp) homotop flows allow vertical composition.
35 Path space preserving homotop flows 16 Definition (Psp homotop flow) A homotop flow on a d-space X is a directed homotop H : X I X with H 0 = id X. A homotop flow is path space preserving (psp) if, for all x and all t I, the maps H(, t) induce homotop equivalences H t (x, ) : T (X ) x T (X ) H(,t) H(x,t), σ H t σ (Psp) homotop flows allow vertical composition. H fits into a diagram of morphisms T (X ) x H() T (X ) x H() T (X ) H(x) H() T (X ) x H(x) x T (X) x H H() T (X) H() H(x) T (X) H() x
36 Localization and quotient categor Extension categor 17 For ever psp homotop flow H and objects (x, ) introduce formal morphisms H : (x, ) (Hx, H) and inverses H 1 with obvious compositions ( zig-zags ) and cancellation rules: (H(x, ), σ ) H = (σ x, H(, )).
37 Localization and quotient categor Extension categor 17 For ever psp homotop flow H and objects (x, ) introduce formal morphisms H : (x, ) (Hx, H) and inverses H 1 with obvious compositions ( zig-zags ) and cancellation rules: (H(x, ), σ ) H = (σ x, H(, )). Compression: From the resulting categor, form a categor (of components) with path components (wrt. isos) as objects. For ever psp homotop flow, (x, ) and (H(x), H()) give rise to the same object most often also (x, H()). Psp homotop flow ensures that objects related b isomorphism have futures and pasts related b isomorphisms. The quotient functor is full faithful and surjective, hence an equivalence of (topologicall enriched) categories.
38 Localization and quotient categor Extension categor 17 For ever psp homotop flow H and objects (x, ) introduce formal morphisms H : (x, ) (Hx, H) and inverses H 1 with obvious compositions ( zig-zags ) and cancellation rules: (H(x, ), σ ) H = (σ x, H(, )). Compression: From the resulting categor, form a categor (of components) with path components (wrt. isos) as objects. For ever psp homotop flow, (x, ) and (H(x), H()) give rise to the same object most often also (x, H()). Psp homotop flow ensures that objects related b isomorphism have futures and pasts related b isomorphisms. The quotient functor is full faithful and surjective, hence an equivalence of (topologicall enriched) categories. Alternative construction (losing information, but universal): Stable components (K. Ziemiański, arxiv , recent)
39 Discretization via subcategor for a pre-cubical set X 18 Pre-cubical set: like pre-simplicial set, composed from cubes. Barcentric subdivision bd(x ). Pick barcenters (one for each cell). Categor with pairs of barcenters as objects. Morphisms: Extensions given b piecewise linear dipaths through barcenters (and composition).
40 Discretization via subcategor for a pre-cubical set X 18 Pre-cubical set: like pre-simplicial set, composed from cubes. Barcentric subdivision bd(x ). Pick barcenters (one for each cell). Categor with pairs of barcenters as objects. Morphisms: Extensions given b piecewise linear dipaths through barcenters (and composition). Result: A categor that is certainl bisimilar to the extension categor also equivalent? (details to be checked). Extends a result b J. Dubut for Euclidean cubical complexes (subdivision not needed).
41 Dir. homotop equivalence f : X Y? Yielding equivalent extension categories 19 Requirements A reverse dimap g : Y X and directed (zig-zag) psp!! dihomotopies H between id X and g f, resp. K between id Y and f g.
42 Dir. homotop equivalence f : X Y? Yielding equivalent extension categories 19 Requirements A reverse dimap g : Y X and directed (zig-zag) psp!! dihomotopies H between id X and g f, resp. K between id Y and f g. Properties Homotop tpes of path spaces are preserved: T (X ) x T (Y ) f fx. in a coherent wa: Equivalent extension categories (+,-) (or their discrete counterparts). 2-out-of-3-propert?
43 Thanks! The end 20 Thanks to the organizers ou, the audience, at the end of a long da of lectures!
44 Thanks! The end 20 Thanks to the organizers ou, the audience, at the end of a long da of lectures!??
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