Ordinal subdivision and special pasting in quasicategories

Ehlers, P.J.; Porter, Timothy

doi:10.1016/j.aim.2007.05.023

Cited by 7 publications

(5 citation statements)

References 22 publications

(27 reference statements)

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“…A similar argument using the inclusion I(n) ֒→ ∆[n] of the spine into the simplex which induces the Segal maps shows that Γ(X) is a Segal space. Now we give a very brief treatment of ordinal subdivisions of simplicial sets and of categories, following work of Ehlers and Porter [EP08].…”

Section: Appendix: Some Results On Quasi-categoriesmentioning

confidence: 99%

Comparison of Waldhausen constructions

Bergner,

Osorno,

Ozornova

et al. 2019

Preprint

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Section: Appendix: Some Results On Quasi-categoriesmentioning

confidence: 99%

Comparison of Waldhausen constructions

Bergner,

Osorno,

Ozornova

et al. 2019

Preprint

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“…Let C be the (O ×G )-cubical set naturally sending each pair (U, g) to C(g). Let H be the (O × G )-stream map X × S I → sd 4 We now prove our main simplicial and cubical approximation theorems.…”

Section: Homotopymentioning

confidence: 98%

“…The last line holds because P [n] is the colimit of all preordered sets [k] [n] , taken over all monotone functions [k] → P . [24], otherwise known as ordinal subdivision [4], plays a role in directed topology analogous to the role barycentric subdivision [3] plays in classical topology. A description [4] of edgewise subdivision in terms of ordinal sums in ∆ makes it convenient for us to reason about double edgewise subdivision [ Lemma 5.10].…”

Section: Simplicial Modelsmentioning

confidence: 99%

“…[24], otherwise known as ordinal subdivision [4], plays a role in directed topology analogous to the role barycentric subdivision [3] plays in classical topology. A description [4] of edgewise subdivision in terms of ordinal sums in ∆ makes it convenient for us to reason about double edgewise subdivision [ Lemma 5.10]. We refer the reader to [4] for more details and Figure 3 for a comparison between edgewise and barycentric subdivisions.…”

Section: Simplicial Modelsmentioning

confidence: 99%

“…A description [4] of edgewise subdivision in terms of ordinal sums in ∆ makes it convenient for us to reason about double edgewise subdivision [ Lemma 5.10]. We refer the reader to [4] for more details and Figure 3 for a comparison between edgewise and barycentric subdivisions.…”

Section: Simplicial Modelsmentioning

confidence: 99%

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Cubical Approximation for Directed Topology I

Krishnan

2013

Appl Categor Struct

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Topological spaces -such as classifying spaces, configuration spaces and spacetimes -often admit extra directionality. Qualitative invariants on such directed spaces often are more informative, yet more difficult, to calculate than classical homotopy invariants because directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Directed spaces often arise as geometric realizations of simplicial sets and cubical sets equipped with ordertheoretic structure encoding the orientations of simplices and 1-cubes. We show that, under definitions of weak equivalences appropriate for the directed setting, geometric realization induces an equivalence between homotopy diagram categories of cubical sets and directed spaces and that its right adjoint satisfies a homotopical analogue of excision. In our directed setting, cubical sets with structure reminiscent of higher categories serve as analogues of Kan complexes. Along the way, we prove appropriate simplicial and cubical approximation theorems and give criteria for two different homotopy relations on directed maps in the literature to coincide.

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