Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
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 , which reproduce the previous case when .
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 for the category of simplicial objects in , hence for the category of bisimplicial sets. This inherits from 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
- projective or injective model structure on functors , .
Write 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 , 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 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).
The model category structure thus obtained is characterized as follows.
The category of bisimplicial sets carries the structure of an sSet-enriched category with hom-object for two bisimplicial sets and given by
This refines to the structure of a
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 :
Let be a simplicial combinatorial model category. Write for the (∞,1)-category presented by it. Write for the functor category/category of simplicial objects in .
(Lurie, prop. 1.5.4, remark 1.5.6)
Cartesian monoidal model structure
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)).
For , write
for the nerve of the free groupoid on (the codiscrete groupoid in objects.) This extends to a functor
We have the following pair of adjoint functors between simplicial sets and bisimplicial sets.
The first is
where sends a bisimplicial set to the simplicial set of its first row
The other is
assigns the total simplicial set to a bisimplicial set: it is the left Kan extension of the functor given by (with from def. 2) along the Yoneda embedding
and where 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).
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
Write for the tree with a single edge and no non-trivial vertex.
Then slice category of over 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
- 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
A survey of the model structures and their relations is in
The generalization to complete Segal objects in model categories other than was considered in
Discussion in terms of Cisinski model structures is in