Weak complicial sets II - nerves of complicial Gray-categories

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Abstract

This paper continues the development of a simplicial theory of weak omega-categories, by studying categories which are enriched in weak complicial sets. These complicial Gray-categories generalise both the Kan complex enriched categories of homotopy theory and the Gray-categories of weak 3-category theory. We derive a simplicial nerve construction, which is closely related to Cordier and Porter's homotopy coherent nerve, and show that this faithfully represents complicial Gray-categories as weak complicial sets. The category of weak complicial sets may itself be canonically enriched to a complicial Gray-category whose homsets are higher generalisations of the bicategory of homomorphisms, strong transformations and modifications. By applying our nerve construction to this structure, we demonstrate that the totality of all (small) weak complicial sets and their structural morphisms at higher dimensions form a richly structured (large) weak complicial set.
LanguageEnglish
Pages441-467
Number of pages27
JournalContemporary mathematics
Volume431
Publication statusPublished - 2007

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Keywords

  • simplicial sets
  • stratification
  • weak complicial set
  • homotopy coherent nerve
  • Gray-category
  • model category
  • enriched categories

Cite this

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Weak complicial sets II - nerves of complicial Gray-categories. / Verity, Dominic.

In: Contemporary mathematics, Vol. 431, 2007, p. 441-467.

Research output: Contribution to journalArticleResearchpeer-review

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PY - 2007

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AB - This paper continues the development of a simplicial theory of weak omega-categories, by studying categories which are enriched in weak complicial sets. These complicial Gray-categories generalise both the Kan complex enriched categories of homotopy theory and the Gray-categories of weak 3-category theory. We derive a simplicial nerve construction, which is closely related to Cordier and Porter's homotopy coherent nerve, and show that this faithfully represents complicial Gray-categories as weak complicial sets. The category of weak complicial sets may itself be canonically enriched to a complicial Gray-category whose homsets are higher generalisations of the bicategory of homomorphisms, strong transformations and modifications. By applying our nerve construction to this structure, we demonstrate that the totality of all (small) weak complicial sets and their structural morphisms at higher dimensions form a richly structured (large) weak complicial set.

KW - simplicial sets

KW - stratification

KW - weak complicial set

KW - homotopy coherent nerve

KW - Gray-category

KW - model category

KW - enriched categories

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SP - 441

EP - 467

JO - Contemporary mathematics

T2 - Contemporary mathematics

JF - Contemporary mathematics

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