nLab
complete Segal space

Idea

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.

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 $X_{•}$ in sSet (a bisimplicial set);

such that

1. there is a composition operation well defined up to coherent homotopy: exibited by the canonical morphisms

   $$X_k\to X_1{\times }_{X_0}\cdots {\times }_{X_0}{X}_1$$X_k \to X_1 \times_{X_0} \cdots \times_{X_0} X_1

   (into the iterated homotopy pullback of the ∞-groupoid of 1-morphisms over the $\mathrm{\infty }$-groupoid of objects) being homotopy equivalences

   (so far this defines a Segal space);

2. the notion of equivalence in $X_{•}$ is compatible with that in the ambient ∞Grpd (“completeness”): the sub-simplicial object $\mathrm{Core}(X_{•})$ 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 $X_0$ ).

Definition

We first discuss

• Complete Segal spaces

as such, and then the more general notion of

• Complete Segal space objects

internal to a suitable model category/$(\mathrm{\infty },1)$-category $𝒞$ – this reduces to the previous notion for $𝒞={\mathrm{sSet}}_{\mathrm{Quillen}}$.

Complete Segal spaces

Definition

A Segal space is a simplicial object in simplicial sets

$$X\in [{\Delta }^{\mathrm{op}},\mathrm{sSet}]$$X \in [\Delta^{op}, sSet]

such that

• it is fibrant in the Reedy model structure $[{\Delta }^{\mathrm{op}},{\mathrm{sSet}}_{\mathrm{Quillen}}{]}_{\mathrm{Reedy}}$;

• it is a local object with respect to the spine inclusions $\{\mathrm{Sp}[n]\hookrightarrow \Delta [n]{\}}_{n\in \mathbb{N}}$;

  equivalently: for all $n\in \mathbb{N}$ the canonical morphism

  $$X_n\to X_1{\times }_{X_0}\cdots {\times }_{X_0}{X}_1$$X_n \to X_1 \times_{X_0} \cdots \times_{X_0} X_1

  is a weak homotopy equivalence (of Kan complexes, in fact).

(Rezk, 4.1).

Definition

For $X$ a Segal space, its homotopy category $\mathrm{Ho}(X)$ is the Ho(Top)-enriched category whose objects are the vertices of $X_0$

$$\mathrm{Obj}(X)=(X_0{)}_0$$Obj(X) = (X_0)_0

and for $x,y\in \mathrm{Obj}(X)$ the hom object is the homotopy type of the homotopy fiber product

$$\mathrm{Ho}(X)(x,y):={\pi }_0(\{x\}{\times }_{X_0}{X}_1{\times }_{X_0}\{y\}).$$Ho(X)(x,y) := \pi_0 \Big(\{x\} \times_{X_0} X_1 \times_{X_0} \{y\}\Big) \,.

The composition

$${\mathrm{Ho}}_X(x,y)\times {\mathrm{Ho}}_X(y,z)\to {\mathrm{Ho}}_X(x,z)$$Ho_X(x,y) \times Ho_X(y,z) \to Ho_X(x,z)

is the (uniquely defined) action of the infinity-anafunctor

$$X_1{\times }_{X_0}{X}_1\underset{\simeq }{\overset{(d_2,d_0)}{\leftarrow }}{X}_2\stackrel{d_1}{\to }{X}_1$$X_1 \times_{X_0} X_1 \underoverset{\simeq}{(d_2, d_0)}{\leftarrow} X_2 \stackrel{d_1}{\to} X_1

on these connected components.

(Rezk, 5.3)

Definition

For $X$ a Segal space, write

$$X_{\mathrm{hoequ}}\hookrightarrow X_1$$X_{hoequ} \hookrightarrow X_1

for the inclusion of the connected components of those vertices that become isomorphisms in the homotopy category, def. 2.

(Rezk, 5.7)

Definition

A Segal space $X$ is called a complete Segal space if

$$s_0:X_0\to X_{\mathrm{hoequ}}$$s_0 : X_0 \to X_{hoequ}

is a weak equivalence.

(Rezk, 6.)

Remark

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.

Remark

The completeness condition may also be thought of as univalence. See there for more.

Complete Segal space objects

(…)

Properties

Characterization of Completeness

Theorem

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

$$X_0\simeq [{\Delta }^{\mathrm{op}},\mathrm{sSet}](*,X)\to [{\Delta }^{\mathrm{op}},\mathrm{sSet}](N(0\stackrel{\simeq }{\to }1),X)$$X_0 \simeq [\Delta^{op}, sSet](*, X) \to [\Delta^{op}, sSet](N(0 \stackrel{\simeq}{\to} 1), X)

is a weak equivalence.

(Rezk, theorem 6.2)

Model category structure

The category $[{\Delta }^{\mathrm{op}},\mathrm{sSet}]$ 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.

Examples

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)

Preliminaries

We need the following basic ingredients.

Write $(-{)}^{(-)}:{\mathrm{Cat}}^{\mathrm{op}}\times \mathrm{Cat}\to \mathrm{Cat}$ for the internal hom in Cat, sending two categories $A$, $X$ to the functor category $X^A=\mathrm{Func}(A,X)$.

By the discussion at nerve we have a canonical functor

$$\Delta \hookrightarrow \mathrm{Cat}$$\Delta \hookrightarrow Cat

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

$$N:\mathrm{Cat}\to \mathrm{sSet}$$N : Cat \to sSet

to sSet sending a category $C$ to

$$N(C):k\mapsto C^{\Delta [k]}.$$N(C) : k \mapsto C^{\Delta[k]} \,.

Its restriction along $\mathrm{Grpd}\hookrightarrow \mathrm{Cat}$ to groupoids lands in Kan complexes $\mathrm{KanCplx}\hookrightarrow$ sSet.

The core operation is the functor

$$\mathrm{Core}:\mathrm{Cat}\to \mathrm{Grpd}$$Core : Cat \to Grpd

right adjoint to the inclusion of Grpd into Cat. It sends a category to the groupoid obtained by discarding all non-invertible morphisms.

The construction

Let $C$ be a small category. Define

$$C\in [{\Delta }^{\mathrm{op}},\mathrm{sSet}]$$\mathbf{C} \in [\Delta^{op}, sSet]

by

$$C_k:=N(\mathrm{Core}(C^{\Delta [k]})).$$\mathbf{C}_k := N(Core(C^{\Delta[k]})) \,.

In degree 0 this is the the core of $C$ itself. In degree 1 it is the groupoid $C_1$ 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) $C_1{\times }_{C_0}\cdots {\times }_{C_0}{C}_1$ is given already be the ordinary pullback in the 1-category Grpd. Using this, it is immediate that for all $k$ the functors

$$\mathrm{Core}(C^{\Delta [k]})\to \mathrm{Core}(C^{\Delta [1]}){\times }_{\mathrm{Core}(C)}\cdots {\times }_{\mathrm{Core}(C)}\mathrm{Core}(C^{\Delta [1]})$$Core(C^{\Delta[k]}) \to Core(C^{\Delta[1]}) \times_{Core(C)} \cdots \times_{Core(C)} Core(C^{\Delta[1]})

are isomorphisms, and so in particular

$$C_k\to C_1{\times }_{C_0}\cdots {\times }_{C_0}{C}_1$$\mathbf{C}_k \to \mathbf{C}_1 \times_{\mathbf{C}_0} \cdots \times_{\mathbf{C}_0} \mathbf{C}_1

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

$$\mathrm{Core}(C):k\mapsto N(\mathrm{Core}(C{)}^{\Delta [k]})=N(\mathrm{Core}(C){)}^{\Delta [k]},$$Core(\mathbf{C}) : k \mapsto N(Core(C)^{\Delta[k]}) = N(Core(C))^{\Delta[k]} \,,

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 $N(\mathrm{Core}(C))$.

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 $C$ constructed as above is a complete Segal space. This construction extends to a functor $\mathrm{Cat}\to \mathrm{completeSegalSpace}$ and this is homotopy full and faithful.

Properties of the inclusion

Write

$${\mathrm{Sing}}_J:\mathrm{Cat}\to [{\Delta }^{\mathrm{op}},\mathrm{sSet}]$$Sing_J : Cat \to [\Delta^{op}, sSet]

for the functor just defined

Proposition

For $C$ and $D$ two categories, there are natural isomorphisms

$${\mathrm{Sing}}_J(C\times D)\simeq {\mathrm{Sing}}_J(C)\times {\mathrm{Sing}}_J(D)$$Sing_J(C \times D) \simeq Sing_J(C) \times Sing_J(D)

and

$${\mathrm{Sing}}_J(D^C)\simeq ({\mathrm{Sing}}_{J}D{)}^{{\mathrm{Sing}}_{J}C}.$$Sing_J(D^C) \simeq (Sing_J D)^{Sing_J C} \,.

A functor $f:C\to D$ is an equivalence of categories precisely if ${\mathrm{Sing}}_J(f)$ is an equivalence in the Reedy model structure $[{\Delta }^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{Reedy}}$ (hence is degreewise a weak homotopy equivalence of Kan complexes).

This appears as (Rezk, theorem 3.7).

Model categories as complete Segal spaces

Let $C$ be a category with a class $W\subset \mathrm{Mor}(C)$ of weak equivalences. For instance, $C$ could be a model category. Then the above construction has the following evident variant.

Definition

Let $N(C,W)\in [{\Delta }^{\mathrm{op}},\mathrm{sSet}]$ be given by

$$N(C,W):n\mapsto N({\mathrm{Core}}_W(C^{\Delta [n]})),$$N(C,W) : n \mapsto N(Core_W(C^{\Delta[n]})) \,,

where now ${\mathrm{Core}}_W(-)$ denotes the subcategory on those natural transformations whose components are weak equivalences in $C$.

Remark

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.

Remark

If $C$ is a model category, then ${\mathrm{Core}}_W(C^{\Delta [n]})$ 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 $N({\mathrm{Core}}_W(C^{\Delta [n]}))$, but only “arranges them more nicely”).

Proposition

Any Reedy fibrant replacement of $N(C,W)$ is a complete Segal space.

This is (Rezk, theorem 8.3).

Quasi-categories as complete Segal spaces

Definition

Write

$${\Delta }_J:\Delta \to \mathrm{sSet}$$\Delta_J : \Delta \to sSet

for the cosimplicial simplicial set that sends $[n]$ to the nerve of the codiscrete groupoid on $n+1$ objects

$${\Delta }_J[n]=N(0\stackrel{\simeq }{\to }\cdots \stackrel{\simeq }{\to }n).$$\Delta_J[n] = N(0 \stackrel{\simeq}{\to} \cdots \stackrel{\simeq}{\to} n) \,.

Write

$${\mathrm{Sing}}_J:\mathrm{sSet}\to [{\Delta }^{\mathrm{op}},\mathrm{sSet}]$$Sing_J : sSet \to [\Delta^{op}, sSet]

for the functor given by

$${\mathrm{Sing}}_J(X{)}_n={\mathrm{Hom}}_{\mathrm{sSet}}(\Delta [n]\times {\Delta }_J[•],X).$$Sing_J(X)_n = Hom_{sSet}(\Delta[n] \times \Delta_J[\bullet], X) \,.

Proposition

For $X\in \mathrm{sSet}$ a quasi-category/inner Kan complex, ${\mathrm{Sing}}_J(X)$ is a complete Segal space.

See at model structure for dendroidal complete Segal spaces the section Quasi-operads to dendroidal complete Segal spaces

Related concepts

• semi-Segal space, Segal space

• model structure for complete Segal spaces

• higher complete Segal space, dendroidal complete Segal space

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

References

General

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

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

A survey of the definition and its relation to equivalent definitions is in section 4 of

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

See also pages 29 to 31 of

• Jacob Lurie, On the Classification of Topological Field Theories

For literature on the variants and refinements see at Theta space and n-fold complete Segal space.

Groupoidal version

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)