category object in an (∞,1)-category, groupoid object
equivalences in/of $(\infty,1)$-categories
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 $X$ is
for each $n \in \mathbb{N}$ a Kan complex $X_n$, thought of as the space of composable sequences of $n$-morphisms and their composites;
forming a simplicial object $X_\bullet$ in sSet (a bisimplicial set);
such that
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 $\infty$-groupoid of objects) being homotopy equivalences
(so far this defines a Segal space);
the notion of equivalence in $X_\bullet$ is compatible with that in the ambient ∞Grpd (“completeness”): the sub-simplicial object $Core(X_\bullet)$ 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 $X_0$ ).
We first discuss
as such, and then the more general notion of
internal to a suitable model category/$(\infty,1)$-category $\mathcal{C}$ – this reduces to the previous notion for $\mathcal{C} = sSet_{Quillen}$.
A Segal space is a simplicial object in simplicial sets
such that
it is fibrant in the Reedy model structure $[\Delta^{op}, sSet_{Quillen}]_{Reedy}$;
it is a local object with respect to the spine inclusions $\{Sp[n] \hookrightarrow \Delta[n]\}_{n \in \mathbb{N}}$;
equivalently: for all $n \in \mathbb{N}$ the Segal map
is a weak homotopy equivalence (of Kan complexes, in fact).
(Rezk, 4.1).
For $X$ a Segal space, its homotopy category $Ho(X)$ is the Ho(Top)-enriched category whose objects are the vertices of $X_0$
and for $x,y \in Obj(X)$ the hom object is the homotopy type of the homotopy fiber product
The composition
is the (uniquely defined) action of the infinity-anafunctor
on these connected components.
For $X$ 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 $X$ is called a complete Segal space if
is a weak equivalence.
(Rezk, 6.)
This condition is equivalent to $X$ being a local object with respect to the morphism $N(\{0 \stackrel{\simeq}{\to} 1\}) \to *$. This is discussed below.
The completeness condition may also be thought of as univalence. See there for more.
(…)
A Segal space $X$ is a complete Segal space precisely if it is a local object with respect to the morphism $N(0 \stackrel{\simeq}{\to} 1) \to *$, hence precisely if with respect to the canonical sSet-enriched hom objects we have that
is a weak equivalence.
The category $[\Delta^{op}, sSet]$ 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.
Given $\mathcal{C}$ a category with weak equivalences $\mathcal{W} \subset Mor(\mathcal{C})$ 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 $N_{Rezk}(\mathcal{C},\mathcal{W})$ .
On the other hand classical simplicial localization-theory provides several ways (e.g. hammock localization) to turn $(\mathcal{C}, \mathcal{W})$ into an (∞,1)-category $\mathcal{C}[\mathcal{W}]^{-1}$ which universally turns the elements in $\mathcal{W}$ 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.
There is a notion of right/left fibration of complete Segal spaces analogous to right/left Kan fibrations? for quasi-categories.
We discuss some examples. For more and more basic examples see also at Segal space – Examples.
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 $(-)^{(-)} : Cat^{op} \times Cat \to Cat$ for the internal hom in Cat, sending two categories $A$, $X$ to the functor category $X^A = Func(A,X)$.
By the discussion at nerve we have a canonical functor
including the simplex category into Cat by regarding the simplex $\Delta[n]$ as the category generated from $n$ consecutive morphisms.
The nerve itself is then then functor
to sSet sending a category $C$ to
Its restriction along $Grpd \hookrightarrow Cat$ to groupoids lands in Kan complexes $KanCplx \hookrightarrow$ 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 $C$ be a small category. Define
by
In degree 0 this is the the core of $C$ itself. In degree 1 it is the groupoid $\mathbf{C}_1$ underlying the arrow category of $C$.
One sees that the source and target functors $s, t : C^{\Delta[1]} \to C$ are isofibrations and hence their image under core and nerve are Kan fibrations. Therefore it follows that the homotopy pullback (see there) $\mathbf{C}_1 \times_{\mathbf{C}_0} \cdots \times_{\mathbf{C}_0} \mathbf{C}_1$ is given already be the ordinary pullback in the 1-category Grpd. Using this, it is immediate that for all $k$ 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 $C$. In particular the morphisms that are invertible under this composition are precisely those that are already invertible in $C$. Therefore we have the core simplicial object
where, note, now we first take the core of $C$ and then form morphism categories.
This simplicial Kan complex has in each positive degree a path space object for the Kan complex $N(Core(C))$.
Notably (since $\Delta[k]$ 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 $C$, the simplicial object $\mathbf{C}$ constructed as above is a complete Segal space. This construction extends to a functor $Cat \to completeSegalSpace$ and this is homotopy full and faithful.
Write
for the functor just defined
For $C$ and $D$ two categories, there are natural isomorphisms
and
A functor $f : C \to D$ is an equivalence of categories precisely if $Sing_J(f)$ is an equivalence in the Reedy model structure $[\Delta^{op}, sSet]_{Reedy}$ (hence is degreewise a weak homotopy equivalence of Kan complexes).
This appears as (Rezk, theorem 3.7).
Let $C$ be a category with a class $W \subset Mor(C)$ of weak equivalences. For instance, $C$ could be a model category or (much) more generally a “relative category”. Then the above construction has the following evident variant.
Let $N(C,W) \in [\Delta^{op}, sSet]$ be given by
where now $Core_W(-)$ denotes the subcategory on those natural transformations whose components are weak equivalences in $C$.
The typical model category is not a small category with respect to the base choice of universe. In this case $N(C,W)$ will be a “large” bisimplicial set. In other words, one needs to employ some universe enlargement to interpret this definition.
If $C$ is a model category, then $Core_W(C^{\Delta[n]})$ is the subcategory of weak equivalences in any of the standard model structures on functors on $C^{\Delta[n]}$. By a classical fact discssed at (∞,1)-categorical hom-space, its nerve is a model for the core of the corresponding (∞,1)-category of (∞,1)-functors.
The bisimplicial set $N(C,W)$ is not, in general, a complete Segal space. It does, however, represent the same (∞,1)-category as the simplicial localization of $C$ at $W$; see this MO question.
We can, of course, always reflect $N(C,W)$ 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 $N(Core_W(C^{\Delta[n]}))$, but only “arranges them more nicely”).
Any Reedy fibrant replacement of $N(C,W)$ is a complete Segal space.
This is (Rezk, theorem 8.3).
Write
for the cosimplicial simplicial set that sends $[n]$ to the nerve of the codiscrete groupoid on $n+1$ objects
Write
for the functor given by
For $X \in sSet$ a quasi-category/inner Kan complex, $Sing_J(X)$ is a complete Segal space.
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
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)