Iterated Chromatic Subdivisions are Collapsible

The standard chromatic subdivision of the standard simplex is a combinatorial algebraic construction, which was introduced in theoretical distributed computing, motivated by the study of the view complex of layered immediate snapshot protocols. A most important property of this construction is the fact that the iterated subdivision of the standard simplex is contractible, implying impossibility results in fault-tolerant distributed computing. Here, we prove this result in a purely combinatorial way, by showing that it is collapsible, studying along the way fundamental combinatorial structures present in the category of colored simplicial complexes.

[1]  S. Lane.  Categories for the Working Mathematician , 1971 .

[2]  Joins for (augmented) simplicial sets , 1999, math/9904039.

[3]  S. Maclane,et al.  Categories for the Working Mathematician , 1971 .

[4]  Maurice Herlihy,et al.  The topological structure of asynchronous computability , 1999, JACM.

[5]  Eric Goubault,et al.  Trace Spaces: An Efficient New Technique for State-Space Reduction , 2012, ESOP.

[6]  J. Whitehead.  Simplicial Spaces, Nuclei and m‐Groups , 1939 .

[7]  Eli Gafni,et al.  Generalized FLP impossibility result for t-resilient asynchronous computations , 1993, STOC.

[8]  Seif Haridi,et al.  Distributed Algorithms , 1992, Lecture Notes in Computer Science.

[9]  Marshall M. Cohen.  A Course in Simple-Homotopy Theory , 1973 .

[10]  Nancy A. Lynch,et al.  Impossibility of distributed consensus with one faulty process , 1985, JACM.

[11]  Dmitry N. Kozlov,et al.  Combinatorial Algebraic Topology , 2007, Algorithms and computation in mathematics.

[12]  Maurice Herlihy,et al.  The asynchronous computability theorem for t-resilient tasks , 1993, STOC.

[13]  Paul G. Goerss,et al.  Simplicial Homotopy Theory , 2009, Modern Birkhäuser Classics.

[14]  Sergio Rajsbaum,et al.  New combinatorial topology bounds for renaming: The upper bound , 2012, JACM.

[15]  Dmitry N. Kozlov,et al.  Topology of the view complex , 2013, ArXiv.

[16]  Michael E. Saks,et al.  Wait-free k-set agreement is impossible: the topology of public knowledge , 1993, STOC.

[17]  Dmitry N. Kozlov,et al.  Chromatic subdivision of a simplicial complex , 2012 .

[18]  Maurice Herlihy,et al.  Distributed Computing Through Combinatorial Topology , 2013 .

[19]  I. Moerdijk,et al.  Sheaves in geometry and logic: a first introduction to topos theory , 1992 .

[20]  M. Grandis.  Directed Algebraic Topology: Models of Non-Reversible Worlds , 2009 .

[21]  Shmuel Zaks,et al.  A combinatorial characterization of the distributed tasks which are solvable in the presence of one faulty processor , 1988, PODC '88.