---
title: Rational Complete Segal Spaces
url: https://www.emergentmind.com/topics/rational-complete-segal-spaces
type: topic
---

# Rational Complete Segal Spaces

Rational complete Segal spaces are one of the two models introduced for rational \((\infty,1)\)-categories in “Models for rational \((\infty, 1)\)-categories” [2509.22413]. In that framework, a rational \((\infty,1)\)-category is an \((\infty,1)\)-category enriched in spaces whose higher homotopy groups \(\pi_n\), for \(n\ge 2\), are rational vector spaces, and a rational complete Segal space is a complete Segal space whose mapping spaces are rational in precisely that sense. The construction is formulated more generally for localization at a multiplicative subset \(M\subset \mathbb{Z}\), with the rational case obtained by taking \(M=\mathbb{Z}\setminus\{0\}\) so that \(M^{-1}\mathbb{Z}=\mathbb{Q}\) [2509.22413].

## 1. Rational \((\infty,1)\)-categories and \(M\)-locality

An ordinary \((\infty,1)\)-category can be regarded as a category enriched in spaces: for each pair of objects \(x,y\), there is a mapping space \(\operatorname{map}(x,y)\), whose points encode 1-morphisms and whose higher homotopy encodes higher morphisms and homotopies. In the rational theory, the enrichment is restricted by a homotopical linearity condition on higher homotopy groups: a space \(Y\) is \(M\)-local if for all \(n\ge 2\), \(\pi_n(Y)\) is an \(M^{-1}\mathbb{Z}\)-module, and \(Y\) is rational when \(M=\mathbb{Z}\setminus\{0\}\), so that \(\pi_n(Y)\) is a rational vector space for every \(n\ge 2\) [2509.22413].

This formulation follows the framework of Gómez-Tato–Halperin–Tanré for rational homotopy of non-simply connected spaces. The resulting theory imposes no restriction on \(\pi_0\) or \(\pi_1\); only the higher groups \(\pi_n\), \(n\ge 2\), are required to be localized. That flexibility is important because mapping spaces in \((\infty,1)\)-categories are often neither connected nor simply connected. A rational complete Segal space therefore models an \((\infty,1)\)-category whose mapping spaces are rational spaces in this non-simply-connected sense [2509.22413].

The motivation parallels classical rational homotopy theory. Rationalization suppresses torsion phenomena and replaces part of the homotopy-theoretic complexity by linear algebra over \(\mathbb{Q}\). In the higher-categorical setting, this supports the study of homotopy-coherent structures up to rational equivalence and provides a model-categorical setting in which rational higher categories can be compared with other presentations of \((\infty,1)\)-categories [2509.22413].

## 2. Complete Segal spaces as the ambient model

A simplicial space is a functor
\[
X\colon \Delta^{\mathrm{op}} \to \mathcal{SSets},
\]
equivalently a bisimplicial set. For a simplicial space \(W\), the Segal maps
\[
\varphi_n\colon W_n \longrightarrow \underbrace{W_1 \underset{W_0}{\times} W_1 \underset{W_0}{\times}\cdots\underset{W_0}{\times} W_1}_{n\text{ factors}}
\]
are induced from the inclusions of the spine \(G(n)\subseteq \Delta[n]\). A Segal space is a Reedy fibrant simplicial space for which all \(\varphi_n\) are weak homotopy equivalences. A complete Segal space is a Segal space satisfying an additional completeness condition expressing that homotopy equivalences are detected at the level of objects [2509.22413].

For completeness, let \(I\) be the category with two objects and a unique isomorphism between them, and \(E=\operatorname{nerve}(I)\). Then for a Segal space \(W\), the map \(E\to \Delta[0]\) induces
\[
W_0 \longrightarrow \operatorname{Map}_{\mathcal{SSets}^{\Delta^{\mathrm{op}}}}(E^{t},W),
\]
and \(W\) is complete when this is a weak homotopy equivalence. Rezk’s model structure \(\mathcal{CSS}\) on simplicial spaces has monomorphisms as cofibrations, complete Segal spaces as fibrant objects, and weak equivalences detected by mapping into complete Segal spaces. Moreover, \(\mathcal{CSS}\) is cartesian [2509.22413].

The relevance of complete Segal spaces to the general theory of \((\infty,1)\)-categories is standard in the literature. Rezk’s nerve construction realizes relative categories with suitable fraction conditions as Segal spaces, and under saturation as complete Segal spaces after Reedy fibrant replacement [1409.8192]. The Yoneda formalism for complete Segal spaces, including the \(\infty\)-category of spaces, left fibrations, and representable presheaves, is likewise available entirely within the complete Segal space model [1401.5656]. Rational complete Segal spaces are built by imposing rationality on mapping spaces inside this established ambient framework.

## 3. Definition of rational complete Segal spaces

If \(W\) is a complete Segal space, its objects are \(\operatorname{ob}(W)=W_{0,0}\). For objects \(x,y\in \operatorname{ob}(W)\), the mapping space \(\operatorname{map}_W(x,y)\) is defined by the pullback
\[
\begin{tikzcd}
\operatorname{map}_W(x,y) \arrow[r] \arrow[d] & W_1 \arrow[d,"(d_1,d_0)"] \\
\Delta[0] \arrow[r,"(x,y)"] & W_0\times W_0.
\end{tikzcd}
\]
A complete Segal space \(W\) is \(M\)-local if, for any two objects \(x,y\), the mapping space \(\operatorname{map}_W(x,y)\) is an \(M\)-local space. It is rational if, for any two objects \(x,y\), \(\operatorname{map}_W(x,y)\) is rational, meaning that \(\pi_n(\operatorname{map}_W(x,y))\) is a \(\mathbb{Q}\)-vector space for all \(n\ge 2\) [2509.22413].

The definition is therefore entirely fiberwise on mapping spaces. Rationality is not imposed on the object space \(W_0\), nor on \(\pi_1\) of the mapping spaces. It is a condition on the higher homotopy of the homotopy-coherent morphism objects. This makes rational complete Segal spaces well adapted to the standard non-simply-connected phenomena of higher category theory [2509.22413].

A plausible implication is that rational complete Segal spaces isolate the “rational part” of higher categorical composition without forcing a collapse of discrete or fundamental-group-level information. That reading is consistent with the paper’s emphasis on the Gómez-Tato–Halperin–Tanré framework and with its use of only the higher homotopy groups \(\pi_n\) for \(n\ge 2\) in the definition of \(M\)-locality [2509.22413].

## 4. Model structure and cartesian property

The model structure for rational complete Segal spaces is obtained by left Bousfield localization of \(\mathcal{CSS}\). For simplicial sets, the paper first constructs a model category \(\mathcal{L}_{M^{-1}\mathbb{Z}}\mathcal{SSets}\) whose fibrant objects are \(M\)-local Kan complexes, using local spheres \(S^n_{M^{-1}\mathbb{Z}}\) and a function-complex characterization of locality. For simplicial spaces, levelwise \(M\)-locality is encoded by the localization set
\[
\widetilde{T}_{M^{-1}\mathbb{Z}}
=
\left\{
(\Delta[i]^t \times \operatorname{Sing}(S^n))_r
\to
(\Delta[i]^t \times \operatorname{Sing}(S^n_{M^{-1}\mathbb{Z}}))_r
\,\middle|\,
i\ge 0,\ n\ge 2
\right\},
\]
where \((-)_r\) is the functor collapsing the 0-simplices to a discrete set [2509.22413].

The resulting theorem states that there is a combinatorial, left proper, simplicial model structure \(\mathcal{L}_{M^{-1}\mathbb{Z}}\mathcal{CSS}\) on simplicial spaces in which weak equivalences are those maps detected by mapping into \(M\)-local complete Segal spaces, cofibrations are monomorphisms, and fibrant objects are exactly the \(M\)-local complete Segal spaces. In the rational case, this yields a model category whose fibrant objects are the rational complete Segal spaces [2509.22413].

A central structural property is that this localized model structure is cartesian. The proof uses Rezk’s criterion and shows that if \(W\) is an \(M\)-local complete Segal space, then the exponential \(W^{\Delta[1]^t}\) is again \(M\)-local. Consequently, \(\mathcal{L}_{M^{-1}\mathbb{Z}}\mathcal{CSS}\), and in particular \(\mathcal{L}_{\mathbb{Q}}\mathcal{CSS}\), is cartesian [2509.22413].

This cartesian property matters because internal homs and exponentials are central to higher-categorical constructions. Related work on complete Segal spaces uses precisely this kind of cartesian and fibrational structure to formulate Yoneda embeddings, representable left fibrations, adjunctions, and limits [1401.5656; 1805.03131]. In the rational setting, the paper’s result shows that rationalization does not destroy the cartesian model-categorical framework needed for those constructions.

## 5. Equivalence with rational Segal categories

The second model introduced in the paper is that of rational Segal categories. A Segal precategory is a simplicial space \(X\) whose \(0\)-space \(X_0\) is discrete, and a Segal category is a Segal precategory whose Segal maps are weak homotopy equivalences. Because \(X_0\) is discrete, one has
\[
X_1=\coprod_{(x,y)}\operatorname{map}_X(x,y),
\]
so \(M\)-locality of mapping spaces is equivalent to levelwise \(M\)-locality [2509.22413].

Using the same localization set \(\widetilde{T}_{M^{-1}\mathbb{Z}}\), the paper constructs a combinatorial, left proper, simplicial model structure \(\mathcal{L}_{M^{-1}\mathbb{Z}}\mathcal{SeCat}_c\) on Segal precategories whose fibrant objects are the Reedy fibrant \(M\)-local Segal categories. In particular, one obtains a model category whose fibrant objects are the Reedy fibrant rational Segal categories [2509.22413].

The main comparison theorem states that for each multiplicative subset \(M\subset \mathbb{Z}\), the adjunction
\[
I\colon \mathcal{L}_{M^{-1}\mathbb{Z}}\mathcal{SeCat}_c
\rightleftarrows
\mathcal{L}_{M^{-1}\mathbb{Z}}\mathcal{CSS}
\colon R
\]
is a Quillen equivalence. Hence the model category for rational complete Segal spaces and the model category for rational Segal categories are Quillen equivalent, and both present the same homotopy theory of rational \((\infty,1)\)-categories [2509.22413].

This equivalence is established by localizing Bergner’s Quillen equivalence between ordinary Segal categories and complete Segal spaces. The strategy uses Hirschhorn’s theorem on compatible Bousfield localizations and the fact that both underlying model categories are combinatorial, left proper, and simplicial, with all objects cofibrant because cofibrations are monomorphisms [2509.22413]. In encyclopedic terms, rational complete Segal spaces are therefore not an isolated construction but one side of a two-model presentation theorem.

## 6. Relation to rational homotopy theory, examples, and significance

The construction is explicitly tied to rational homotopy theory through local spheres \(S^n_{M^{-1}\mathbb{Z}}\), built as mapping telescopes of degree maps
\[
S^n \xrightarrow{m_1} S^n \xrightarrow{m_2} S^n \to \cdots
\]
for \(m_i\in M\). A space \(Y\) is \(M\)-local if and only if for every \(n\ge 2\), the restriction
\[
\operatorname{Map}_{\mathcal{Top}}(S^n_{M^{-1}\mathbb{Z}},Y)\to \operatorname{Map}_{\mathcal{Top}}(S^n,Y)
\]
is a weak homotopy equivalence. The paper transfers this characterization from topological spaces to simplicial sets and then to simplicial spaces via the representables \(\Delta[i]^t\) [2509.22413].

The approach is purely model-categorical. The paper does not directly use Sullivan cdga models or Quillen dg Lie models, although it states that its constructions are compatible with such algebraic models in principle. A plausible implication is that rational complete Segal spaces provide a homotopy-coherent presentation of rational higher categories that can later be compared with algebraic models of rational homotopy when desired [2509.22413].

Explicit examples are limited, but the paper notes that spaces such as \(S^1\), \(S^1\vee S^1\), and \(\mathbb{R}P^\infty\) have vanishing higher homotopy groups and hence are automatically rational. Therefore any \((\infty,1)\)-category whose mapping spaces are of this form is automatically rational. More generally, by choosing \(M\) as the complement of a set of primes \(P\subset\mathbb{Z}\), one obtains \(P\)-local \((\infty,1)\)-categories as a direct generalization of rational ones [2509.22413].

The broader significance claimed in the paper is foundational. Rational complete Segal spaces and rational Segal categories provide explicit, equivalent models for rational \((\infty,1)\)-categories. The cartesian property of the rational complete Segal space model makes it particularly suitable for internal homs, exponentials, and enrichment, while the discrete-object model of rational Segal categories gives a more combinatorial presentation. The paper also identifies future directions, notably rational analogs of quasi-categories and simplicial categories, with the aim of extending the equivalence of models beyond the complete Segal space and Segal category settings [2509.22413].

Source: https://www.emergentmind.com/topics/rational-complete-segal-spaces