nLab model structure for complete Segal spaces

Idea

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.

Definition

We first discuss the model structure

• for complete Segal spaces

as such, and then more generally model structures

• for complete Segal space objects

in a suitable ambient model category $\mathcal{C}$, which reproduce the previous case when $\mathcal{C} = sSet_{Quillen}$.

For complete Segal spaces

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

• Reedy model structure$[\Delta^{op}, sSet]_{Reedy}$,

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

• projective or injective model structure on functors $[\Delta^{op}, sSet]_{proj}$, $[\Delta^{op}, sSet]_{inj}$.

Write $c : Set \to sSet$ for the inclusion of sets as discrete objects into simplicial sets. Write

$$[\Delta^{op}, c] : sSet \to [\Delta^{op}, sSet]$$

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!).

Definition

The model structure for complete Segal spaces is the left Bousfield localization of $[\Delta^{op}, sSet]_{Reedy}$ at the set of morphisms

$$W := \{ Sp[n] \hookrightarrow \Delta[n] \}_{n \in \mathbb{N}} \cup \{ N(\{a \stackrel{\simeq}{\to} b\}) \to * \} \,,$$

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).

Remark

Equivalently one can replace here the spine inclusions with the inner horn inclusions.

Remark

An object $X \in [\Delta^{op}, sSet]$ being a local object with respect to the $n$th spine inclusion says that the morphism

$$X_n \to X_1 \times_{X_0} \cdots \times_{X_0} X_1$$

(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.

Proposition

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

$$hom(X,Y) := i_2^*(Y^X) \,,$$

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

• left proper

• cartesianclosed monoidal

• simplicial model category

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).

For complete Segal space objects

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 $C$.

Proposition

There is the structure of a simplicial model category on $[\Delta^{op}, C]$ such that

1. is is a left Bousfield localization of the injective model structure on functors $[\Delta^{op}, C]_{inj}$;

2. 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)

Properties

Cartesian monoidal model structure

Proposition

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).

Relation to other model structures

Model structure for quasi-categories

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)).

Definition

For $n \in \mathbb{N}$, write

$$\Delta_J[n] := N(0 \stackrel{\simeq}{\to} \cdots \stackrel{\simeq}{\to} n) \in sSet$$

for the nerve of the free groupoid on $\Delta[n]$ (the codiscrete groupoid in $(n+1)$ objects.) This extends to a functor

$$\Delta_J : \Delta \to sSet \,.$$

Proposition

We have the following pair of adjoint functors between simplicial sets and bisimplicial sets.

The first is

$$(p_1^* \dashv i_1^*) : [\Delta^{op}, Set] \stackrel{\overset{i_1^*}{\leftarrow}}{\underset{p_1^*}{\to}} [\Delta^{op} \times \Delta^{op}, Set]$$

where $i_1^*$ sends a bisimplicial set to the simplicial set of its first row

$$i_1^* : X_{\bullet,\bullet} \mapsto X_{\bullet, 0} \,.$$

The other is

$$(t_! \dashv t^!) : [\Delta^{op} \times \Delta^{op}, Set] \stackrel{\overset{t^!}{\leftarrow}}{\underset{t_!}{\to}} [\Delta^{op}, Set]$$

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. ) along the Yoneda embedding

  $$t_!(X_{\bullet,\bullet}) = \int^{[k],[l]} X_{k,l} \cdot \Delta[k] \times \Delta_J[l] \,;$$

• and where $t^!$ forms in each degree the mapping space

  $$t^! : X_\bullet \mapsto \left( [m,n] \mapsto Hom_{sSet}( \Delta[m] \times \Delta_J[n], X) \right) \,.$$

Proposition

The composite of the two adjunctions from prop.

$$sSet \stackrel{\overset{i_1^* t^!}{\leftarrow}}{\underset{t_! p_1^*}{\to}} sSet$$

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).

Proposition

Both adjunctions of prop. 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. .

This appears (Joyal-Tierney, theorem 4.11, 4.12).

Model structure for dendroidal complete Segal spaces

The operadic analog of simplicial sets / bisimplicial sets are dendroidal sets / dendroidal spaces, parameterized over the tree category $\Omega$

$$dsSet := [\Omega^{op}, sSet] \,.$$

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

$$ssSet \simeq dsSet_{/\eta} \hookrightarrow dsSet \,.$$

By restriction along this inclusion, the model structure for dendroidal complete Segal spaces reproduces the above model structure. See there for more details.

Related concepts

• model structure for dendroidal complete Segal spaces

• table - models for (infinity,1)-operads

References

Complete Segal spaces were originally defined in

• Charles Rezk, A model for the homotopy theory of homotopy theory , Trans. Amer. Math. Soc., 353(3), 973-1007 (pdf)

The Quillen equivalence with the model structure for quasi-categories is discussed in

• André Joyal, Myles Tierney, Quasi-categories vs Segal spaces (arXiv:0607820)

A survey of the model structures and their relations is in

• Julia Bergner, A survey of $(\infty, 1)$-categories (arXiv).

The generalization to complete Segal objects in model categories other than $sSet$ was considered in

• Jacob Lurie, $(\infty,2)$-Categories and the Goodwillie calculus (arXiv:0905.0462)

Discussion in terms of Cisinski model structures is in

• Dimitri Ara, Higher quasi-categories vs higher Rezk spaces (arXiv:1206.4354)

A model structure for (infinity,2)-sheaves of complete Segal spaces is discussed in

• Nicholas Meadows, Local Complete Segal Spaces (arXiv:1607.05794)