# nLab complete Segal space

## Internal $n$-category

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

# Contents

## 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. They are also called Rezk categories after Charles Rezk.

More in detail, a complete Segal space $X$ is

• for each $n\in ℕ$ 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 Segal maps

${X}_{k}\to {X}_{1}{×}_{{X}_{0}}\cdots {×}_{{X}_{0}}{X}_{1}$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 $\infty$-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 $\mathrm{Core}\left({X}_{•}\right)$ 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

as such, and then the more general notion of

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

### Complete Segal spaces

###### Definition
$X\in \left[{\Delta }^{\mathrm{op}},\mathrm{sSet}\right]$X \in [\Delta^{op}, sSet]

such that

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

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

equivalently: for all $n\in ℕ$ the Segal map

${X}_{n}\to {X}_{1}{×}_{{X}_{0}}\cdots {×}_{{X}_{0}}{X}_{1}$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 $\mathrm{Ho}\left(X\right)$ is the Ho(Top)-enriched category whose objects are the vertices of ${X}_{0}$

$\mathrm{Obj}\left(X\right)=\left({X}_{0}{\right)}_{0}$Obj(X) = (X_0)_0

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

$\mathrm{Ho}\left(X\right)\left(x,y\right):={\pi }_{0}\left(\left\{x\right\}{×}_{{X}_{0}}{X}_{1}{×}_{{X}_{0}}\left\{y\right\}\right)\phantom{\rule{thinmathspace}{0ex}}.$Ho(X)(x,y) := \pi_0 \Big(\{x\} \times_{X_0} X_1 \times_{X_0} \{y\}\Big) \,.

The composition

${\mathrm{Ho}}_{X}\left(x,y\right)×{\mathrm{Ho}}_{X}\left(y,z\right)\to {\mathrm{Ho}}_{X}\left(x,z\right)$Ho_X(x,y) \times Ho_X(y,z) \to Ho_X(x,z)

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

${X}_{1}{×}_{{X}_{0}}{X}_{1}\underset{\simeq }{\overset{\left({d}_{2},{d}_{0}\right)}{←}}{X}_{2}\stackrel{{d}_{1}}{\to }{X}_{1}$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.

###### Definition

For $X$ a Segal space, write

${X}_{\mathrm{hoequ}}↪{X}_{1}$X_{hoequ} \hookrightarrow X_1

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

###### 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 $N\left(\left\{0\stackrel{\simeq }{\to }1\right\}\right)\to *$. This is discussed below.

###### Remark

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

(…)

## 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 $N\left(0\stackrel{\simeq }{\to }1\right)\to *$, hence precisely if with respect to the canonical sSet-enriched hom objects we have that

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

is a weak equivalence.

### Model category structure

The category $\left[{\Delta }^{\mathrm{op}},\mathrm{sSet}\right]$ 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 $\left(-{\right)}^{\left(-\right)}:{\mathrm{Cat}}^{\mathrm{op}}×\mathrm{Cat}\to \mathrm{Cat}$ for the internal hom in Cat, sending two categories $A$, $X$ to the functor category ${X}^{A}=\mathrm{Func}\left(A,X\right)$.

By the discussion at nerve we have a canonical functor

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

including the simplex category into Cat by regarding the simplex $\Delta \left[n\right]$ 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

$N\left(C\right):k↦{C}^{\Delta \left[k\right]}\phantom{\rule{thinmathspace}{0ex}}.$N(C) : k \mapsto C^{\Delta[k]} \,.

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

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

by

${C}_{k}:=N\left(\mathrm{Core}\left({C}^{\Delta \left[k\right]}\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $s,t:{C}^{\Delta \left[1\right]}\to C$ are isofibrations and hence their image under core and nerve are Kan fibrations. Therefore it follows that the homotopy pullback (see there) ${C}_{1}{×}_{{C}_{0}}\cdots {×}_{{C}_{0}}{C}_{1}$ is given already be the ordinary pullback in the 1-category Grpd. Using this, it is immediate that for all $k$ the functors

$\mathrm{Core}\left({C}^{\Delta \left[k\right]}\right)\to \mathrm{Core}\left({C}^{\Delta \left[1\right]}\right){×}_{\mathrm{Core}\left(C\right)}\cdots {×}_{\mathrm{Core}\left(C\right)}\mathrm{Core}\left({C}^{\Delta \left[1\right]}\right)$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

${C}_{k}\to {C}_{1}{×}_{{C}_{0}}\cdots {×}_{{C}_{0}}{C}_{1}$\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

$\mathrm{Core}\left(C\right):k↦N\left(\mathrm{Core}\left(C{\right)}^{\Delta \left[k\right]}\right)=N\left(\mathrm{Core}\left(C\right){\right)}^{\Delta \left[k\right]}\phantom{\rule{thinmathspace}{0ex}},$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 $N\left(\mathrm{Core}\left(C\right)\right)$.

Notably (since $\Delta \left[k\right]$ 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

${\mathrm{Sing}}_{J}:\mathrm{Cat}\to \left[{\Delta }^{\mathrm{op}},\mathrm{sSet}\right]$Sing_J : Cat \to [\Delta^{op}, sSet]

for the functor just defined

###### Proposition

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

${\mathrm{Sing}}_{J}\left(C×D\right)\simeq {\mathrm{Sing}}_{J}\left(C\right)×{\mathrm{Sing}}_{J}\left(D\right)$Sing_J(C \times D) \simeq Sing_J(C) \times Sing_J(D)

and

${\mathrm{Sing}}_{J}\left({D}^{C}\right)\simeq \left({\mathrm{Sing}}_{J}D{\right)}^{{\mathrm{Sing}}_{J}C}\phantom{\rule{thinmathspace}{0ex}}.$Sing_J(D^C) \simeq (Sing_J D)^{Sing_J C} \,.

A functor $f:C\to D$ is an equivalence of categories precisely if ${\mathrm{Sing}}_{J}\left(f\right)$ is an equivalence in the Reedy model structure $\left[{\Delta }^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{Reedy}}$ (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 $W\subset \mathrm{Mor}\left(C\right)$ of weak equivalences. For instance, $C$ could be a model category. Then the above construction has the following evident variant.

###### Definition

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

$N\left(C,W\right):n↦N\left({\mathrm{Core}}_{W}\left({C}^{\Delta \left[n\right]}\right)\right)\phantom{\rule{thinmathspace}{0ex}},$N(C,W) : n \mapsto N(Core_W(C^{\Delta[n]})) \,,

where now ${\mathrm{Core}}_{W}\left(-\right)$ 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 $N\left(C,W\right)$ 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 ${\mathrm{Core}}_{W}\left({C}^{\Delta \left[n\right]}\right)$ is the subcategory of weak equivalences in any of the standard model structures on functors on ${C}^{\Delta \left[n\right]}$. 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 $N\left(C,W\right)$ 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 $N\left(C,W\right)$ 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 $N\left({\mathrm{Core}}_{W}\left({C}^{\Delta \left[n\right]}\right)\right)$, but only “arranges them more nicely”).

###### Proposition

Any Reedy fibrant replacement of $N\left(C,W\right)$ 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 $\left[n\right]$ to the nerve of the codiscrete groupoid on $n+1$ objects

${\Delta }_{J}\left[n\right]=N\left(0\stackrel{\simeq }{\to }\cdots \stackrel{\simeq }{\to }n\right)\phantom{\rule{thinmathspace}{0ex}}.$\Delta_J[n] = N(0 \stackrel{\simeq}{\to} \cdots \stackrel{\simeq}{\to} n) \,.

Write

${\mathrm{Sing}}_{J}:\mathrm{sSet}\to \left[{\Delta }^{\mathrm{op}},\mathrm{sSet}\right]$Sing_J : sSet \to [\Delta^{op}, sSet]

for the functor given by

${\mathrm{Sing}}_{J}\left(X{\right)}_{n}={\mathrm{Hom}}_{\mathrm{sSet}}\left(\Delta \left[n\right]×{\Delta }_{J}\left[•\right],X\right)\phantom{\rule{thinmathspace}{0ex}}.$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, ${\mathrm{Sing}}_{J}\left(X\right)$ is a complete Segal space.

## 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

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

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