A complete Segal space is a model for an internal category in an (∞,1)-category in ∞Grpd, with the latter presented by sSet/Top. So complete Segal spaces present (∞,1)-categories. They are also called Rezk categories after Charles Rezk.
More in detail, a complete Segal space is
there is a composition operation well defined up to coherent homotopy: exibited by the Segal maps
(into the iterated homotopy pullback of the ∞-groupoid of 1-morphisms over the -groupoid of objects) being homotopy equivalences
(so far this defines a Segal space);
the notion of equivalence in is compatible with that in the ambient ∞Grpd (“completeness”): the sub-simplicial object on the invertible morphisms in each degree is homotopy constant: it has all face and degeneracy maps being homotopy equivalences.
(this says that if a morphism is an equivalence under the explicit composition operation then it is already a morphism in ).
We first discuss
as such, and then the more general notion of
internal to a suitable model category/-category – this reduces to the previous notion for .
Complete Segal spaces
For a Segal space, its homotopy category is the Ho(Top)-enriched category whose objects are the vertices of
and for the hom object is the homotopy type of the homotopy fiber product
is the (uniquely defined) action of the infinity-anafunctor
on these connected components.
For a Segal space, write
for the inclusion of the connected components of those vertices that become isomorphisms in the homotopy category, def. 2.
A Segal space is called a complete Segal space if
is a weak equivalence.
Complete Segal space objects
Characterization of Completeness
A Segal space is a complete Segal space precisely if it is a local object with respect to the morphism , hence precisely if with respect to the canonical sSet-enriched hom objects we have that
is a weak equivalence.
(Rezk, theorem 6.2)
Model category structure
The category of simplicial presheaves on the simplex category (bisimplicial sets) supports a model category structure whose fibrant objects are precisely the complete Segal spaces: the model structure for complete Segal spaces. This presents the (∞,1)-category of (∞,1)-categories.
Relation to simplicial localization
Given a category with weak equivalences or more generally a “relative category”, then there is canonically a complete Segal space associated with it by the “relative nerve” construction, def. 2 followed by fibrant replacement, and hence by the model structure this determines an (∞,1)-category .
On the other hand classical simplicial localization-theory provides several ways (e.g. hammock localization) to turn into an (∞,1)-category which universally turns the elements in into homotopy equivalences.
These constructions are compatible in that there is an equivalence of (∞,1)-categories
For simplicial model categories this is (Rezk, theorem 8.3. For general model categories this is (Bergner 07, theorem 6.2). For the fully general case this follows from results by Clark Barwick, Daniel Kan and Bertrand Toën as pointed out by Chris Schommer-Pries here on MathOverflow.
We discuss some examples. For more and more basic examples see also at Segal space – Examples.
Ordinary categories as complete Segal spaces
We discuss how an ordinary small category is naturally regarded as a complete Segal space. (Rezk, 3.5)
We need the following basic ingredients.
Write for the internal hom in Cat, sending two categories , to the functor category .
By the discussion at nerve we have a canonical functor
including the simplex category into Cat by regarding the simplex as the category generated from consecutive morphisms.
The nerve itself is then then functor
to sSet sending a category to
Its restriction along to groupoids lands in Kan complexes sSet.
The core operation is the functor
right adjoint to the inclusion of Grpd into Cat. It sends a category to the groupoid obtained by discarding all non-invertible morphisms.
Let be a small category. Define
In degree 0 this is the the core of itself. In degree 1 it is the groupoid underlying the arrow category of .
One sees that the source and target functors are isofibrations and hence their image under core and nerve are Kan fibrations. Therefore it follows that the homotopy pullback (see there) is given already be the ordinary pullback in the 1-category Grpd. Using this, it is immediate that for all the functors
are isomorphisms, and so in particular
is an equivalence.
It is clear that the composition operation in the complete Segal space defined this way “is” the composition in . In particular the morphisms that are invertible under this composition are precisely those that are already invertible in . Therefore we have the core simplicial object
where, note, now we first take the core of and then form morphism categories.
This simplicial Kan complex has in each positive degree a path space object for the Kan complex .
Notably (since is weak homotopy equivalent to the point) it follows that indeed all the face and degeneracy maps are weak homotopy equivalences.
So for every category , the simplicial object constructed as above is a complete Segal space. This construction extends to a functor and this is homotopy full and faithful.
Properties of the inclusion
for the functor just defined
For and two categories, there are natural isomorphisms
A functor is an equivalence of categories precisely if is an equivalence in the Reedy model structure (hence is degreewise a weak homotopy equivalence of Kan complexes).
This appears as (Rezk, theorem 3.7).
Relative and Model categories as complete Segal spaces
Let be a category with a class of weak equivalences. For instance, could be a model category or (much) more generally a “relative category”. Then the above construction has the following evident variant.
Let be given by
where now denotes the subcategory on those natural transformations whose components are weak equivalences in .
The bisimplicial set is not, in general, a complete Segal space. It does, however, represent the same (∞,1)-category as the simplicial localization of at ; see this MO question.
We can, of course, always reflect into a complete Segal space by passing to a fibrant replacement in the model structure for complete Segal spaces. But something better is true here: it suffices to make a Reedy fibrant replacement (which does not change the homotopy type of the simplicial sets , but only “arranges them more nicely”).
This is (Rezk, theorem 8.3).
Quasi-categories as complete Segal spaces
for the cosimplicial simplicial set that sends to the nerve of the codiscrete groupoid on objects
for the functor given by
See at model structure for dendroidal complete Segal spaces the section Quasi-operads to dendroidal complete Segal spaces
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 relation to quasi-categories is discussed in
Further discussion of the relation to simplicial localization is in
A survey of the definition and its relation to equivalent definitions is in section 4 of
See also pages 29 to 31 of
For literature on the variants and refinements see at Theta space and n-fold complete Segal space.
Related MathOverflow discussion includes
The groupoidal version of complete Segal spaces (that modelling just groupoid objects in an (∞,1)-category instead of general category objects in an (∞,1)-category) is discussed in
Julia Bergner, Adding inverses to diagrams encoding algebraic structures, Homology, Homotopy Appl. 10(2), 2008, 149-174 (arXiv:math/0610291)
Julia Bergner, Adding inverses to diagrams II: Invertible homotopy theories are spaces, Homology, Homotopy and Applications, vol. 10(1) 2008 (pdf)