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

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 $Str \omega Cat \stackrel{resolve}{\to} Trimble \omega Cat$.

Section 6 of

- Dominic Verity,
*Weak complicial sets, a simplicial weak omega-category theory. Part I: basic homotopy theory*(arXiv:math/0604414).

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