# nLab model structure for weak complicial sets

**model category** ## Definitions ## * category with weak equivalences * weak factorization system * homotopy * homotopy category * small object argument * resolution ## Morphisms ## * Quillen adjunction * Quillen equivalence * Quillen bifunctor * derived functor ## Universal constructions ## * homotopy Kan extension * homotopy limit/homotopy colimit * Bousfield-Kan map ## Refinements ## * monoidal model category * monoidal Quillen adjunction * enriched model category * enriched Quillen adjunction * simplicial model category * simplicial Quillen adjunction * cofibrantly generated model category * combinatorial model category * cellular model category * algebraic model category * compactly generated model category * proper model category * cartesian closed model category, locally cartesian closed model category * stable model category ## Producing new model structures * on functor categories (global) * Reedy model structure * on overcategories * Bousfield localization * transferred model structure * model structure on algebraic fibrant objects ## Presentation of $(\infty,1)$-categories ## * (∞,1)-category * simplicial localization * (∞,1)-categorical hom-space * presentable (∞,1)-category ## Model structures ## * Cisinski model structure ### for $\infty$-groupoids for ∞-groupoids * on topological spaces * Strom model structure * Thomason model structure * model structure on presheaves over a test category * on simplicial sets, on semi-simplicial sets * for right/left fibrations * model structure on simplicial groupoids * on cubical sets * on strict ω-groupoids, on groupoids * on chain complexes/model structure on cosimplicial abelian groups related by the Dold-Kan correspondence * model structure on cosimplicial simplicial sets ### for $n$-groupoids * for n-groupoids/for n-types * for 1-groupoids ### for $\infty$-groups * model structure on simplicial groups * model structure on reduced simplicial sets ### for $\infty$-algebras #### general * on monoids * on simplicial T-algebras, on homotopy T-algebras * on algebas over a monad * on algebras over an operad, on modules over an algebra over an operad #### specific * on dg-algebras over an operad * on dg-algebras and on on simplicial rings/on cosimplicial rings related by the monoidal Dold-Kan correspondence * for L-∞ algebras: on dg-Lie algebras, on dg-coalgebras, on simplicial Lie algebras * model structure on dg-modules ### for stable/spectrum objects * model structure on spectra * model structure on presheaves of spectra ### for $(\infty,1)$-categories * on categories with weak equivalences * Joyal model for quasi-categories * on sSet-categories * for complete Segal spaces * for Cartesian fibrations ### for stable $(\infty,1)$-categories * on dg-categories ### for $(\infty,1)$-operads * on operads, for Segal operads on algebras over an operad, on modules over an algebra over an operad * on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations ### for $(n,r)$-categories * for (n,r)-categories as Θ-spaces * for weak ω-categories as weak complicial sets * on cellular sets * on higher categories in general * on strict ω-categories ### for $(\infty,1)$-sheaves / $\infty$-stacks * on homotopical presheaves * on simplicial presheaves global model structure/Cech model structure/local model structure on simplicial sheaves on presheaves of simplicial groupoids on sSet-enriched presheaves * model structure for (2,1)-sheaves/for stacks

# Contents

## Idea

The notion of weak complicial set is a model for (the omega-nerve of) the notion of weak ω-category. The model structure for weak $\omega$-categories is a model category structure on the category of stratified simplicial sets, such that cofibrant-fibrant objects are precisely the weak complicial sets. This model structure may therefore be understood as a presentation of the (∞,1)-category of weak $\omega$-categories.

Urs Schreiber: It would be interesting to see which algebraic definitions of higher categories coul be equipped with a model category structure induced from this model structure for weak complicial sets under the corresponding natural nerve and realization adjunction, maybe even as a transferred model structure.

For instance I imagine that there is a good cosimplicial Trimble ω-category

$\Delta \stackrel{O}{\to} Str \omega Cat \stackrel{resolve}{\to} Trimble \omega Cat \,,$

where the first step is Street’s orientals and the second step is an embedding that regards a strict ω-category as a suitably resolved equivalent Trimble $\omega$-category.

The induced nerve and realization adjunction

$Trimble \omega Cat \stackrel{\overset{|-|}{\leftarrow}}{\underset{N}{\to}} sSet$

might maybe even be used to define the transferred model structure on Trimble $\omega$-catgeories (but I haven’t checked at all if the relevant conditions have a chance of being satisfied, though). In any case I would expect that the counit of the adjunction $|N(C)| \to C$ serves as a good (cofibrant) replacement in $Trimble \omega Cat$ for reasonable choices of $Str \omega Cat \stackrel{resolve}{\to} Trimble \omega Cat$.

## References

Section 6 of

Revised on April 6, 2010 14:32:36 by Urs Schreiber (131.211.235.55)