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
category object in an (∞,1)-category, groupoid object
The notion of complete Segal space is a model for the notion of (∞,1)-category regarded as an internal category in an (∞,1)-category in ∞Grpd.
The model category structure on the collection of all bisimplicial sets is a presentation the (∞,1)-category of all complete Segal spaces, hence for the (∞,1)-category of (∞,1)-categories.
We first discuss the model structure
as such, and then more generally model structures
in a suitable ambient model category $\mathcal{C}$, which reproduce the previous case when $\mathcal{C} = sSet_{Quillen}$.
Write sSet for the category of simplicial sets, which here we always think of as equipped with the standard model structure on simplicial sets that is a presentation of the (∞,1)-category ∞Grpd.
Write $[\Delta^{op}, sSet]$ for the category of simplicial objects in $sSet$, hence for the category of bisimplicial sets. This inherits from $sSet$ in particular the
which is a presentation of the (∞,1)-category of simplicial objects in ∞Grpd. In all of the following in its place one can also use the
Write $c : Set \to sSet$ for the inclusion of sets as discrete objects into simplicial sets. Write
for the corresponding inclusion of simplicial sets into bisimplicial sets.
When we think in the following of a simplicial set in the context of $[\Delta^{op}, sSet]$, we always do so via this inclusion (and not via the other natural such inclusion!).
The model structure for complete Segal spaces is the left Bousfield localization of $[\Delta^{op}, sSet]_{Reedy}$ at the set of morphisms
where the first summand is the set of spine inclusions, while the second summand is the singleton containing the morphism from the nerve of the groupoid with two objects and precisely one non-identity morphism (and its inverse) from one to the other.
This appears originally in section 12 of (Rezk).
An object $X \in [\Delta^{op}, sSet]$ being a local object with respect to the $n$th spine inclusion says that the morphism
(with $n$ factors on the right) is a weak homotopy equivalence of simplicial sets. Therefore the objects which are local with respect to all spine inclusions are precisely the Segal spaces.
Accordingly, an object is furthermore local also with respect to $J \to *$ if it is a complete Segal space.
The model category structure thus obtained is characterized as follows.
The category $[\Delta^{op}, sSet] = [\Delta^{op}\times \Delta^{op}, Set]$ of bisimplicial sets carries the structure of an sSet-enriched category with hom-object for two bisimplicial sets $X$ and $Y$ given by
where
$Y^X$ is the internal hom in the standard closed monoidal structure on presheaves;
$i_2 : \Delta \to \Delta \times \Delta$ is the right adjoint to the projection $\Delta \times \Delta \to \Delta$ on the second factor, given by $i_2 : [n] \mapsto ([0],[n])$.
This refines to the structure of a
by setting
cofibrations are the monomorphisms
weak equivalences are the Rezk weak equivalences:. those morphisms $u : A \to B$ such that for all complete Segal spaces $X$ the morphism $hom(u,X) : hom(B,X) \to hom(A,X)$ is a weak equivalence in the standard model structure on simplicial sets.
The fibrant objects in the structure are precisely the complete Segal spaces.
This is essentially (Rezk, theorem 7.2). See also (Joyal-Tierney, theorem 4.1).
We may generalize from complete Segal spaces to complete Segal space objects in an ambient context other tham ∞Grpd $\simeq (sSet_{Quillen})^\circ$:
Let $C$ be a simplicial combinatorial model category. Write $C^\circ$ for the (∞,1)-category presented by it. Write $[\Delta^{op}, C]$ for the functor category/category of simplicial objects in $C$.
There is the structure of a simplicial model category on $[\Delta^{op}, C]$ such that
is is a left Bousfield localization of the injective model structure on functors $[\Delta^{op}, C]_{inj}$;
such that the fibrant objects $X \colon \Delta^{op} \to C$ are precisely the injectively fibrant objects such that furthermore there image under $[\Delta^{op}, C] \to ([\Delta^{op},C])^\circ$ is an internal (∞,1)-category in $C^\circ$.
(Lurie, prop. 1.5.4, remark 1.5.6)
The category $[\Delta^{op}, sSet]$ is a cartesian closed category. This closed monoidal category-structure is compatible with the model category structure in that it makes $[\Delta^{op}, sSet]_{cSegal}$ into a monoidal model category.
This is the last clause of (Rezk, theorem 7.2). The key lemma for establishing this clause is (Rezk, prop. 9.2).
We discuss the relation to the model structure for quasi-categories.
(See also at model structure for dendroidal complete Segal spaces the section Relation to quasi-operads .)
A quick way to say the following turns out to be to say that the model structure for complete Segal spaces is the simplicial completion of Cisinski model structures of the model structure for quasi-categories (see (Ara)).
For $n \in \mathbb{N}$, write
for the nerve of the free groupoid on $\Delta[n]$ (the codiscrete groupoid in $(n+1)$ objects.) This extends to a functor
We have the following pair of adjoint functors between simplicial sets and bisimplicial sets.
The first is
where $i_1^*$ sends a bisimplicial set to the simplicial set of its first row
The other is
where
$t_!$ assigns the total simplicial set to a bisimplicial set: it is the left Kan extension of the functor $t : \Delta \times \Delta \to ssSet$ given by $([k],[l]) \mapsto \Delta[k]\times \Delta_J[l]$ (with $J$ from def. 2) along the Yoneda embedding
and where $t^!$ forms in each degree the mapping space
The composite of the two adjunctions from prop. 4
is the identity adjunction: both functors are isomorphic to the identity functor.
This appears at the end of the proof of (Joyal-Tierney, theorem 4.12).
Both adjunctions of prop. 4 are Quillen equivalences between the model structure for quasi-categories on simplicial sets and the Rezk model structure for complete Segal spaces on bisimplicial sets, def. 1.
This appears (Joyal-Tierney, theorem 4.11, 4.12).
The operadic analog of simplicial sets / bisimplicial sets are dendroidal sets / dendroidal spaces, parameterized over the tree category $\Omega$
Write $\eta \in \Omega \hookrightarrow dSet \hookrightarrow dsSet$ for the tree with a single edge and no non-trivial vertex.
Then slice category of $dsSet$ over $\eta$ is evidently equivalent to that of bisimplicial sets
By restriction along this inclusion, the model structure for dendroidal complete Segal spaces reproduces the above model structure. See there for more details.
Complete Segal spaces were originally defined in
The Quillen equivalence with the model structure for quasi-categories is discussed in
A survey of the model structures and their relations is in
The generalization to complete Segal objects in model categories other than $sSet$ was considered in
Discussion in terms of Cisinski model structures is in