nLab
Segal space

Idea

A Segal space is a pre-category object in ∞Grpd.

A genuine category object in ∞Grpd is a complete Segal space. This is a way of speaking of (∞,1)-categories.

Definition

A Segal space $X_{•}$ is a simplicial topological space or bisimplicial set $X_{•}:{\Delta }^{\mathrm{op}}\to \mathrm{Top}$ which satisfies the Segal conditions:

for all $m,n\in \mathbb{N}$ the square

$$\begin{array}{ccc}{X}_{m+n}& \stackrel{p_{0,\cdots ,m}^{*}}{\to }& X_m\\ {}^{p_{m,\cdots ,m+n}^{*}}\downarrow & & {\downarrow }^{p_m^{*}}\\ X_n& \stackrel{p_0^{*}}{\to }& X_0\end{array}$$\array{ X_{m+n} &\stackrel{ p^*_{0,\cdots, m} }{\to}& X_m \\ {}^{\mathllap{p^*_{m, \cdots, m+n}}}\downarrow && \downarrow^{\mathrlap{p^*_m}} \\ X_n &\stackrel{p^*_0}{\to}& X_0 }

is a homotopy pullback square.

A Segal space for which $X_0$ is a discrete space is called a Segal category. See there for more dicussion.

Examples

In $\mathrm{Set}$

For $𝒞$ a (small) category we may regard its ordinary nerve simplicial set $N(𝒞)\in {\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}$ as a Segal space, under the canonical inclusion $\mathrm{Set}\hookrightarrow \mathrm{\infty }\mathrm{Grpd}$,

$$N(𝒞)\in {\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}\hookrightarrow \mathrm{\infty }{\mathrm{Grpd}}^{{\Delta }^{\mathrm{op}}}.$$N(\mathcal{C}) \in Set^{\Delta^{op}} \hookrightarrow \infty Grpd^{\Delta^{op}} \,.

In fact, the classical “nerve theorem” about the Segal conditions says that a simplicial set is the nerve of a category precisely if it is a Segal space.

Notice that $\mathrm{Equiv}(N(𝒞{)}_1)\hookrightarrow N(𝒞{)}_1$ is precisely the subset of isomorphisms in all morphisms of $𝒞$.

Therefore under this identification, $N(𝒞)$ is a complete Segal space precisely if $𝒞$ is a gaunt category, hence precisely if the only isomorphisms in $𝒞$ are the identities.

In particular if $𝒞$ is a (0,1)-category, hence a preordered set, then $N(𝒞)$ is complete Segal precisely if $𝒞$ is in fact an partially ordered set.

Construction in $1\mathrm{Grpd}$ from a category

Let $𝒞$ be an ordinary category. We discuss how Segal spaces are associated with this.

Let $𝒦$ be a groupoid and $p:𝒦\to 𝒞$ a functor which is essentially surjective.

Then let $X_1≔(p/p)\in \mathrm{Grpd}$ be the “lax fiber product” of $p$ with itself, or rather the comma object of $p$ with itself, hence the comma category, sitting in the universal square

$$\begin{array}{ccc}{X}_1& \stackrel{{\partial }_1}{\to }& 𝒦\\ {}^{{\partial }_0}\downarrow & ⇙& {\downarrow }^p\\ 𝒦& \underset{p}{\to }& 𝒞\end{array}.$$\array{ X_1 &\stackrel{\partial_1}{\to}& \mathcal{K} \\ {}^{\mathllap{\partial_0}}\downarrow &\swArrow& \downarrow^{\mathrlap{p}} \\ \mathcal{K} &\underset{p}{\to}& \mathcal{C} } \,.

Next let $X_2=(p/p/p)$ be the “3-fold comma category”, hence the comma category in

$$\begin{array}{ccc}{X}_2& \stackrel{{\partial }_0}{\to }& 𝒦\\ {}^{}\downarrow & ⇙& {\downarrow }^p\\ X_1& \underset{{\partial }_1}{\to }& 𝒞\end{array}$$\array{ X_2 &\stackrel{\partial_{0}}{\to}& \mathcal{K} \\ {}^{\mathllap{}}\downarrow &\swArrow& \downarrow^{\mathrlap{p}} \\ X_1 &\underset{\partial_1}{\to}& \mathcal{C} }

and so forth: $X_n≔p^{{/}^{n+1}}$.

This way an object of $X_n$ is an $(n+1)$-tuple of objects $(x_0,x_1,\cdots ,x_n)\in 𝒦$ together with a sequence of $n$ composable morphisms $p(x_0)\to p(x_1)\to \cdots \to p(x_n)$, and a morphism is an $(n+1)$-tuple of morphisms $(f_0,f_1,\cdots ,f_n)\in 𝒦$ and a pasting commuting diagram

$$\begin{array}{ccccccc}p(x_0)& \to & p(x_1)& \to & \cdots & \to & p(x_1)\\ {}^{p(f_0)}\downarrow & & {\downarrow }^{p(f_1)}& & \cdots & & {\downarrow }^{p(f_1)}\\ p(x{\prime }_0)& \to & p(x_1)& \to & \cdots & \to & p(x{\prime }_1)\end{array}$$\array{ p(x_0) &\to& p(x_1) &\to& \cdots &\to& p(x_1) \\ {}^{\mathllap{p(f_0)}}\downarrow && \downarrow^{\mathrlap{p(f_1)}} && \cdots && \downarrow^{\mathrlap{p(f_1)}} \\ p(x'_0) &\to& p(x_1) &\to& \cdots &\to& p(x'_1) }

in $𝒞$.

By direct inspection, the maps $X_n\to X^{\partial {\Delta }^n}$ obtained this way are isofibrations, hence fibrations in the canonical model structure on Grpd and so the homotopy pullbacks that enter the Segal conditions for $X_{•}$ are given by ordinary fiber products. These clearly satisfy the Segal conditions, hence

$$X_{•}\in {\mathrm{Grpd}}^{\Delta }\hookrightarrow \mathrm{\infty }{\mathrm{Grpd}}^{{\Delta }^{\mathrm{op}}}$$X_\bullet \in Grpd^{\Delta} \hookrightarrow \infty Grpd^{\Delta^{op}}

constructed this way is a Segal space.

Two special case of the functor $p$ are important:

• if $𝒦\simeq \mathrm{core}(𝒞)$ is the core of $𝒞$ and $p$ is the canonical core inclusion, one finds that $\mathrm{Equiv}(X_1)\hookrightarrow X_1$ by the above construction is $\mathrm{Equiv}(X_1)=\mathrm{Core}(𝒞{)}^{{\Delta }^1}$, the arrow category of the core of $𝒞$. This is equivalent to $𝒞$ by, for instance, the source or restriction map. Hence for $p$ the core inclusion, the above construction gives the complete Segal space corresponding to the category $𝒞$.

• if $p:{\pi }_0(𝒞)\to 𝒞$ is a choice of basepoints in each isomorphism class of $𝒞$, then $X_{•}$ is the Segal category incarnation of the category $𝒞$.

In $1\mathrm{Grpd}$

We consider the situation of From a category, but now conversely, starting with a Segal space in groupoids and then extracting a category from it.

Consider a Segal space that is degreewise just a 1-groupoid, hence a simplicial object in the inclusion

$$X_{•}\in 1{\mathrm{Grpd}}^{{\Delta }^{\mathrm{op}}}\hookrightarrow \mathrm{\infty }{\mathrm{Grpd}}^{{\Delta }^{\mathrm{op}}}.$$X_\bullet \in 1Grpd^{\Delta^{op}} \hookrightarrow \infty Grpd^{\Delta^{op}} \,.

Choosing this to be Reedy fibrant, the map $({\partial }_0,{\partial }_1):X_1\to X_0\times X_0$ is an isofibration.

We may write an object $K\in X_1$ as a horizontal morphism

$${\partial }_1(K)\stackrel{K}{\to }{\partial }_2(K)$$\partial_1(K) \stackrel{K}{\to} \partial_2(K)

and a morphism $\lambda :L\to K$ in $X_1$ as a vertical double category arrow:

$$\begin{array}{ccc}{\partial }_1(L)& \stackrel{L}{\to }& {\partial }_0(L)\\ {}^{{\partial }_1(\lambda )}\downarrow & {\Downarrow }^{\lambda }& {\downarrow }^{{\partial }_0(\lambda )}\\ {\partial }_1(K)& \stackrel{K}{\to }& {\partial }_0(K)\end{array}.$$\array{ \partial_1(L) & \stackrel{L}{\to} & \partial_0(L) \\ {}^{\mathllap{\partial_1(\lambda)}}\downarrow &\Downarrow^{\mathrlap{\lambda}}& \downarrow^{\mathrlap{\partial_0(\lambda)}} \\ \partial_1(K) &\stackrel{K}{\to}& \partial_0(K) } \,.

Then the fact that $({\partial }_1,{\partial }_0)$ is an isofibration means that for every “niche”

$$\begin{array}{ccc}{y}_0& & y_1\\ {}^{f_0}\downarrow & & {\downarrow }^{f_1}\\ x_0& \stackrel{K}{\to }& x_1\end{array},$$\array{ y_0 & & y_1 \\ {}^{\mathllap{f_0}}\downarrow && \downarrow^{\mathrlap{f_1}} \\ x_0 &\stackrel{K}{\to}& x_1 } \,,

namely for every pair of morphisms $f_0,f_1$ in $X_0$ and lift of its codomain to an object $K\in X_1$, there is a “niche filler”

$$\begin{array}{ccc}{y}_0& \stackrel{L}{\to }& Y_1\\ {}^{f_0}\downarrow & {\Downarrow }^{\lambda }& {\downarrow }^{f_1}\\ x_0& \stackrel{K}{\to }& x_1\end{array},$$\array{ y_0 & \stackrel{L}{\to} & Y_1 \\ {}^{\mathllap{f_0}}\downarrow &\Downarrow^{\mathrlap{\lambda}}& \downarrow^{\mathrlap{f_1}} \\ x_0 &\stackrel{K}{\to}& x_1 } \,,

namely a lift of the whole pair $(f_0,f_1)$ to a morphism $\lambda$ in $X_1$, and this is necessarily universal in that any other such lift uniquely factors through this one (because $X_1$ is a groupoid).

Comparison with the definition of a 2-category equipped with proarrows in the incarnation as a double category shows that this is the beginning of the construction of a pseudo double category whose vertical category is $X_0$ and whose weak horizontal composition is that induced by the Segal maps.

Assume next that $X_3\to X^{\partial {\Delta }^2}$ is a 1-monomorphism, as are all the higher $X^n\to X^{\partial {\Delta }^n}$, for $n\ge 3$, hence that $X_{•}$ is 2-coskeletal as a simplicial object. This means that the horizontal composition in this pseudo double category has unique composites, hence that the horizontal category is an ordinary category. If then furthermore the composite $\mathrm{Equiv}(X_1)\to X_1\stackrel{{\partial }_0}{\to }{X}_0$ is an equivalence, hence is the Segal space is a complete Segal space this means that $X_{•}$ arises from this horizontal category by the construction above.

Related notions

• reduced Segal space, Segal category,

• semi-Segal space

• complete Segal space, model structure for complete Segal spaces

References

See generally the references at complete Segal space.

The “Segal conditions” are first discussed in

• Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math., vol. 34, pp. 105–112 (1968),

where it is attributed to Alexander Grothendieck.

The term “Segal space” is due to

• Charles Rezk, …

The invertible case of Segal spaces, hence models for groupoid objects in an (infinity,1)-category are discussed in section 3 of

• Julia Bergner, Adding inverses to diagrams II: Invertible homotopy theories are spaces, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. (web, arXiv:0710.2254)