on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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
symmetric monoidal (∞,1)-category of spectra
The operadic generalization of the model structure on sSet-categories: a presentation of (∞,1)Operad.
All operads considered here are multi-coloured symmetric operads (symmetric multicategories).
Call a morphism of simplicial operads $f : P \to Q$
fully faithful if it is a weak homotopy equivalence on all component simplicial sets.
essentially surjective if the induced morphism $\pi_0(f)$ on homotopy categories is an essentially surjective functor.
a weak equivalence if it is fully faithful and essentially surjective.
a local fibration if it is componentwise a Kan fibration.
a fibration if it is a local fibration and the underlying functor $\pi_0(f) : Ho(j^* P) \to Ho(j^* Q)$ on the homotopy categories of the underlying simplicial categories is an isofibration.
This defines on $sSet Operad$ the structure of a model category which is
This is (Cisinski-Moerdijk, theorem 1.14).
For $C \in$ Set, let $sSet Operad_C \hookrightarrow sSet Operad$ be the full subcategory on operads with $C$ as their set of colours.
Then $sSet Operad_C \simeq (Operad_C)^{\Delta^{op}}$ is the category of simplicial objects in $C$-coloured symmetric operads, and restricted to this the above model category structure is corresponding the model structure on simplicial algebras.
See (Cisinski-Moerdijk, remark 1.9).
Restricted along the inclusion
the above model structure restricts to the model structure on sSet-categories by Julie Bergner.
A morphism in $sSet Operad$ is an acyclic fibration precisely if it is componentwise an acyclic Kan fibration.
The entries of the following table display models, model categories, and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third table) and of $\mathcal{O}$-monoidal (∞,1)-categories (fourth table).
general pattern | ||||
---|---|---|---|---|
strict enrichment | (∞,1)-category/(∞,1)-operad | |||
$\downarrow$ | $\downarrow$ | |||
enriched (∞,1)-category | $\hookrightarrow$ | internal (∞,1)-category | ||
(∞,1)Cat | ||||
SimplicialCategories | $-$homotopy coherent nerve$\to$ | SimplicialSets/quasi-categories | RelativeSimplicialSets | |
$\downarrow$simplicial nerve | $\downarrow$ | |||
SegalCategories | $\hookrightarrow$ | CompleteSegalSpaces | ||
(∞,1)Operad | ||||
SimplicialOperads | $-$homotopy coherent dendroidal nerve$\to$ | DendroidalSets | RelativeDendroidalSets | |
$\downarrow$dendroidal nerve | $\downarrow$ | |||
SegalOperads | $\hookrightarrow$ | DendroidalCompleteSegalSpaces | ||
$\mathcal{O}$Mon(∞,1)Cat | ||||
DendroidalCartesianFibrations |