Simplicial structures on model categories and functors

By Charles Rezk, S. Schwede, B. Shipley. 2001

84 citations recorded by the source corpus.

We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or "continuous," functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0fc5c3c44919d009bb8306b0966d1ff16aabb050

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A Cellular Nerve for Higher Categories (2002, methodology): “The localisation of the Reedy model structure for simplicial spaces (i.e., bisimplicial sets) has been studied by Dugger [16] and Rezk– Schwede–Shipley [32].”
The 2-category theory of quasi-categories (2013, unknown): “Adjunctions in homotopical contexts are commonly presented as Quillen adjunctions, which can be replaced by adjunctions of this type in good set-theoretical cases [22].”
On differential graded categories (2006, methodology): “A general method to prove that a triangulated category obtained from a suitable stable Quillen model category is not algebraic is to show that its [123] Hom-functor enriched in spectra does not factor through the canonical functor from the derived category of abelian groups to the homotopy category ”
Diagram spaces and symmetric spectra (2011, background): “In [25] the above lemma is only stated for simplicial sets, but the analogous result for (compactly generated weak Hausdorff) topological spaces is an immediate consequence.”
Stable model categories are categories of modules (2003, background): “This is not a big loss of generality; it is shown in [46] that every cofibrantly generated, proper, stable model category is in fact Quillen equivalent to a simplicial model category.”
Higher Topos Theory (2006, background): “There is an analogous assertion for model categories but it is more complicated: any suitably nice model category is equivalent to a model category enriched over the category Set∆ of simplicial sets, in an essentially unique way: see [43] for a precise statement and a proof.”
A universal characterization of higher algebraic K-theory (2010, methodology): “By applying the techniques of [12, 48] (as in the proof of corollary 2.”
Morita theory in abelian, derived and stable model categories (2003, background): “model category; that construction does however not yield symmetric spectra, and there is no good composition pairing. The condition of being a simplicial model category can be removed by appealing to [RSS] or [Dug] where suitable model categories are replaced by Quillen equivalent simplicial model c”