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D. M. Kan

Publications and source records attributed to D. M. Kan.

8 recordsLinked to original sources
arXiv2013

Partial model categories and their simplicial nerves

In this note we consider partial model categories, by which we mean relative categories that satisfy a weakened version of the model category axioms involving only the weak equivalences. More precisely, a partial model category will be a relative category that has the two out of six property and admits a 3-arrow calculus. We then show that Charles Rezk's result that the simplicial space obtained from a simplicial model category by taking a Reedy fibrant replacement of its simplicial nerve is a complete Segal space also holds for these partial model categories. We also note that conversely every complete Segal space is Reedy equivalent to the simplicial nerve of a partial model category and in fact of a homotopically full subcategory of a category of diagrams of simplicial sets.

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arXiv2013

Quillen Theorems Bn for homotopy pullbacks of (infinity, k)-categories

We extend the Quillen Theorem Bn for homotopy fibers of Dwyer, et al. to similar results for homotopy pullbacks and note that these results imply similar results for zigzags in the categories of relative categories and k-relative categories, not only with respect to their Reedy structures but also their Rezk structure, which turns them into models for the theories of (infinity, 1)- and (infinity, k)-categories, respectively.

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arXiv2011

n-Relative Categories

We introduce, for every positive integer n, the notion of an n-relative category and show that the category of the small n-relative categories is a model for the homotopy theory of n-fold homotopy theories, i.e. homotopy theories of ... of homotopy theories.

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arXiv2011

A Quillen theorem Bn for homotopy pullbacks

We prove an extension of the Quillen Theorem Bn for homotopy fibres to a similar result for homotopy pullbacks and use this to obtain sufficient conditions on a pullback diagram of categories to guarantee that it be a homotopy pullback.

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arXiv2011

A characterization of simplicial localization functors

We characterize simplicial localization functors among relative functors from relative categories to simplicial categories as any choice of homotopy inverse to the delocalization functor of Dwyer and the second author.

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arXiv2011

In the category of relative categories the Rezk equivalences are exactly the DK-equivalences

In a previous paper we lifted Charles Rezk's complete Segal model structure on the category of simplicial spaces to a Quillen equivalent one on the category of "relative categories," and our aim in this successor paper is to obtain a more explicit description of the weak equivalences in this model structure by showing that these weak equivalences are exactly the DK-equivalences, i.e. those maps between relative categories which induce a weak equivalence between their simplicial localizations.

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