---
title: Rational Segal Categories Overview
url: https://www.emergentmind.com/topics/rational-segal-categories
type: topic
---

# Rational Segal Categories Overview

Searching arXiv for the cited papers and closely related work on rational Segal categories.
{"query": "\"Segal Enriched Categories I\" Bacard 1009.3673 rational Segal categories", "max_results": 5}
{"query": "\"A new model of dg-categories\" 2308.06417 dg-Segal spaces", "max_results": 5}
{"query": "\"Models for rational (\\infty, 1)-categories\" rational Segal categories 2509.22413", "max_results": 5}
Rational Segal categories arise in several closely related frameworks that combine Segal-type composition laws with rational or $\mathbb{Q}$-linear structure. In the sense of Segal enriched categories, they are path-objects $F:P_X\to M$ valued in a rational base such as $\mathrm{Vect}_{\mathbb{Q}}$ or $\mathrm{Ch}_{\mathbb{Q}}$, with composition encoded by Segal colaxity maps and controlled by a chosen class $W$ of homotopy equivalences [1009.3673]. In a dg-homotopical setting, complete dg-Segal spaces over $\mathbb{Q}$ provide a model for dg-categories enriched in rational chain complexes [2308.06417]. In a more recent $(\infty,1)$-categorical usage, rational Segal categories are Segal precategories whose mapping spaces are rational spaces, equivalently fibrant objects in a left Bousfield localization of Bergner’s Segal-category model structure, and they are Quillen equivalent to rational complete Segal spaces [2509.22413]. The term therefore does not denote a single universal definition; rather, it refers to a family of Segal-style models in which rationality is imposed either by linear enrichment over $\mathbb{Q}$ or by localization at rational equivalences.

## 1. Foundational Segal-enriched formalism

The basic ambient datum is a base $(M,W)$, where $M$ is a bicategory and $W$ is a class of 2-cells called homotopy equivalences. The class $W$ is required to contain all invertible 2-cells, satisfy vertical 3-out-of-2, and be stable under horizontal composition. For a small category $C$, the associated strict 2-category $P_C$ is the 2-path-category: objects are those of $C$, 1-morphisms are finite composable strings in $C$, and 2-morphisms are generated by compositions and insertions of identities. Concatenation of chains defines horizontal composition in $P_C$ [1009.3673].

A path-object of $(M,W)$ is a $W$-colax morphism of bicategories
\[
F:P_C\to M
\]
whose colaxity 2-cells, called Segal maps, lie in $W$. Explicitly, for $A,B,C\in C$ and composable strings $(t,s)$ one has
\[
F_{AC}(t\otimes s)\xrightarrow{\varphi(A,B,C)(t,s)}F_{BC}(t)\otimes F_{AB}(s)\in W,
\]
and for each $A$,
\[
F_{AA}([0,A])\xrightarrow{\varphi_A}I'_{FA}\in W.
\]
When $C=X$ is the coarse category on a nonempty set $X$, such a path-object is called a Segal $M_W$-category, or an $X$-point of $(M,W)$ [1009.3673].

This formalism packages object sets externally and hom-objects internally. The objects are the elements of $X$, while the hom-object from $x$ to $y$ is determined by $F([1,(x,y)])$. Composition is induced by concatenation in the path-bicategory together with the Segal maps, so associativity and unitality hold strictly only when the relevant maps are isomorphisms, and otherwise hold up to the chosen class $W$ [1009.3673].

In cartesian settings, this reproduces the standard simplicial Segal condition. Colax monoidal functors $\Delta\to M$ correspond to simplicial objects, and the classical Segal maps take the form
\[
s_n:X_n\to X_1\times_{X_0}\cdots\times_{X_0}X_1,\qquad n\ge 2.
\]
The enriched formalism replaces those fiber products by tensor products of hom-objects, which is the mechanism that makes linear and dg variants possible [1009.3673].

## 2. Rational linear and differential graded realizations

In the strict rational linear case, one takes
\[
M=(\mathrm{Vect}_{\mathbb{Q}},\otimes_{\mathbb{Q}},\mathbb{Q}),\qquad W=\mathrm{Iso}(M).
\]
A Segal $\mathrm{Vect}_{\mathbb{Q}}$-category on an object set $X$ is then an $X$-point $F:P_X\to \mathrm{Vect}_{\mathbb{Q}}$ whose Segal maps are isomorphisms. This is equivalent to a classical $\mathbb{Q}$-linear enriched category with
\[
\operatorname{Map}_C(x,y):=F_{xy}([1,(x,y)])\in \mathrm{Vect}_{\mathbb{Q}},
\]
composition
\[
\circ:\operatorname{Map}_C(y,z)\otimes_{\mathbb{Q}}\operatorname{Map}_C(x,y)\to \operatorname{Map}_C(x,z),
\]
and unit
\[
\mathbb{Q}\to \operatorname{Map}_C(x,x).
\]
Because the Segal condition is strict here, associativity and unitality hold on the nose [1009.3673].

The dg-rational case replaces vector spaces by chain complexes:
\[
M=(\mathrm{Ch}_{\mathbb{Q}},\otimes_{\mathbb{Q}},\mathbb{Q}),\qquad W=\text{quasi-isomorphisms}.
\]
A Segal DG-category is an $X$-point $F:P_X\to \mathrm{Ch}_{\mathbb{Q}}$ such that all colaxity maps are quasi-isomorphisms:
\[
F_{xz}(t\otimes s)\xrightarrow{\varphi(x,y,z)(t,s)}F_{yz}(t)\otimes_{\mathbb{Q}}F_{xy}(s),
\]
and
\[
F_{xx}([0,x])\xrightarrow{\varphi_x}\mathbb{Q}.
\]
Its mapping objects are complexes $\operatorname{Map}_C(x,y)\in \mathrm{Ch}_{\mathbb{Q}}$, and composition is given by a chain map obtained as the composite
\[
\operatorname{Map}_C(y,z)\otimes \operatorname{Map}_C(x,y)\xleftarrow[\sim]{\varphi^{-1}}F_{xz}([2,(x,y,z)])\xrightarrow{}F_{xz}([1,(x,z)]).
\]
Units are induced by $[0,x]\to [1,(x,x)]$ and $\varphi_x^{-1}$ [1009.3673].

In this dg setting, associativity and unitality hold up to $W$, namely in the homotopy category $\mathrm{Ho}(\mathrm{Ch}_{\mathbb{Q}})$. The differential is compatible with tensor products and composition; for homogeneous $\alpha,\beta$,
\[
d(\beta\otimes \alpha)=d\beta\otimes \alpha+(-1)^{|\beta|}\beta\otimes d\alpha,
\]
and
\[
d(\beta\circ \alpha)=d\beta\circ \alpha+(-1)^{|\beta|}\beta\circ \alpha.
\]
Over a general commutative ring $R$, the same paper notes that one should use chain homotopy equivalences rather than quasi-isomorphisms, since quasi-isomorphisms are not necessarily stable under $\otimes_R$; over $\mathbb{Q}$ and, more generally, over fields, quasi-isomorphisms are admissible [1009.3673].

## 3. Role of the Segal maps and the class \(W\)

The class $W$ is the homotopical control mechanism in Segal enrichment. Requiring the colaxity maps and unit maps to lie in $W$ means that composition and identity data are not forced to be strictly invertible, but only invertible up to the specified notion of equivalence. In particular, the maps
\[
\varphi(A,B,C)(t,s):F_{AC}(t\otimes s)\to F_{BC}(t)\otimes F_{AB}(s)
\]
and
\[
\varphi_A:F_{AA}([0,A])\to I'_{FA}
\]
are the enriched analogues of the usual Segal maps, and their membership in $W$ is the Segal condition in the noncartesian setting [1009.3673].

This reformulation is significant because it avoids reliance on fiber products. In the classical cartesian case, Segal conditions are stated using iterated pullbacks over $X_0$. In the enriched path-object case, fiber products are replaced by tensor products, and coherence is shifted into the colaxity data. A plausible implication is that the enriched formalism is especially well-suited to linear and monoidal situations where tensor products, rather than finite products, are the natural compositional operation.

Morphisms of Segal $M_W$-categories are also organized bicategorically. An $M$-premorphism from $F:P_C\to M$ to $G:P_D\to M$ is a pair $(\Sigma,\sigma)$ with a functor $\Sigma:C\to D$ and a transformation of colax morphisms $\sigma:F\Rightarrow G\circ P_\Sigma$. An $M$-morphism is an $M$-premorphism with identity object-components. Equivalences may be formulated strictly when the transformation components are invertible and the induced map on objects is bijective, or homotopically after localization with respect to $W$ [1009.3673].

The same paper provides a canonical reduction process. For any base $(M,W)$ there exists a bicategory $W^{-1}M$ and a homomorphism
\[
L_W:M\to W^{-1}M
\]
that inverts $W$ universally. Consequently, a Segal point $(C,F)$ can be reduced to a strict Segal point $(C,L_W\circ F)$ in the localized base. Concretely, each hom-category $M(U,V)$ is localized at $W_{UV}$, and composition is induced via the universal property of product localizations [1009.3673].

## 4. Rational Segal categories as models for rational \((\infty,1)\)-categories

A later use of the term treats rational Segal categories as Segal-category models for rational $(\infty,1)$-categories. Here rationality is imposed not by enrichment over $\mathbb{Q}$-vector spaces but by requiring mapping spaces to be rational spaces, meaning that for every $n\ge 2$ the homotopy group $\pi_n(Y)$ is a $\mathbb{Q}$-vector space. No constraint is imposed on $\pi_0$ or $\pi_1$. Rationality is encoded model-categorically by localizing simplicial sets at the maps
\[
T_{\mathbb{Q}}=\left\{\operatorname{Sing}(S^n)\to \operatorname{Sing}(S^n_{\mathbb{Q}})\mid n\ge 2\right\},
\]
producing a model category whose fibrant objects are precisely the $\mathbb{Q}$-local Kan complexes and whose weak equivalences are $\mathbb{Q}$-local equivalences [2509.22413].

In this framework, a rational Segal category is a Segal precategory $X$ with discrete $X_0$ such that each mapping space $\operatorname{map}_X(x,y)$ is rational, equivalently such that the simplicial space is levelwise rational. The Segal maps
\[
\varphi_n:X_n\longrightarrow X_1\times_{X_0}\cdots\times_{X_0}X_1,\qquad n\ge 2,
\]
are required to be $\mathbb{Q}$-local weak equivalences [2509.22413].

The corresponding model structure is obtained by left Bousfield localization of Bergner’s $\mathcal{S}\mathrm{e}\mathcal{C}\mathrm{at}_c$ at the set
\[
\widetilde{T}_{\mathbb{Q}}=\left\{(\Delta[i]^t\times \operatorname{Sing}(S^n))_r\to (\Delta[i]^t\times \operatorname{Sing}(S^n_{\mathbb{Q}}))_r\mid i\ge 0,\ n\ge 2\right\}.
\]
The resulting model category
\[
\mathcal{L}_{\mathbb{Q}\mathcal{S}\mathrm{e}\mathcal{C}\mathrm{at}_c}
\]
has monomorphisms as cofibrations, $\mathbb{Q}$-local equivalences as weak equivalences, and Reedy fibrant $\mathbb{Q}$-local Segal categories as fibrant objects [2509.22413].

A rational Dwyer–Kan equivalence between fibrant objects is defined by two conditions: for all objects $x,y$, the induced map
\[
\operatorname{map}_X(x,y)\to \operatorname{map}_Y(fx,fy)
\]
is a $\mathbb{Q}$-local weak homotopy equivalence, and the induced functor on homotopy categories $\mathrm{Ho}(X)\to \mathrm{Ho}(Y)$ is an equivalence of categories. The same paper proves that the classical Quillen equivalence between Segal categories and complete Segal spaces descends to the rational localizations, yielding
\[
I:\mathcal{L}_{\mathbb{Q}\mathcal{S}\mathrm{e}\mathcal{C}\mathrm{at}_c}\simeq_{\text{Q.E.}}\mathcal{L}_{\mathbb{Q}\mathcal{CSS}}:R.
\]
Thus rational Segal categories and rational complete Segal spaces are equivalent homotopy-theoretic models for rational $(\infty,1)$-categories [2509.22413].

## 5. dg-Segal spaces over \(\mathbb{Q}\)

A distinct but adjacent construction models dg-categories by dg-Segal spaces. Fix a commutative ring or field $k$. A dg-category $T$ over $k$ consists of a set of objects, cochain complexes $\operatorname{Hom}_T(x,y)\in C(k)$, composition maps
\[
\operatorname{Hom}_T(y,z)\otimes \operatorname{Hom}_T(x,y)\to \operatorname{Hom}_T(x,z),
\]
and units $e_x:k\to \operatorname{Hom}_T(x,x)$. The paper constructs free dg-categories of finite type and lets $\mathcal{L}$ be the full simplicial subcategory of the simplicial localization whose objects are cofibrant free dg-categories of finite type [2308.06417].

A dg-Segal space is a simplicial functor
\[
F\in \mathrm{Fun}^S(\mathcal{L}^{op},s\mathrm{Set})
\]
satisfying three conditions: coproducts are sent to products up to weak equivalence, $F(\varnothing)\simeq *$, and for every graph-killing construction $G[\overset{\circ}{\alpha}]$ the induced square
\[
F(L(G[\overset{\circ}{\alpha}]))\to F(L(G))
\]
\[
\downarrow\qquad\qquad\downarrow
\]
\[
F(\widehat{c}(1,n,1))\to F((1,n,1))
\]
is a homotopy pullback in simplicial sets. A linearization
\[
j:\Delta\to \mathcal{L}
\]
produces classical Segal maps
\[
F(\widehat{n})\to F(\widehat{1})\times_{F(k)}\cdots\times_{F(k)}F(\widehat{1}),
\]
which are weak equivalences for $n\ge 2$ [2308.06417].

Completeness is encoded by the map
\[
F(k)\to F_{hoequiv},
\]
where $F_{hoequiv}\subset F(\widehat{1})$ is the subspace of 0-simplices giving homotopy equivalences. A complete dg-Segal space is a dg-Segal space satisfying this completeness equivalence. After a second left Bousfield localization, the fibrant objects are precisely the projective-fibrant complete dg-Segal spaces [2308.06417].

When specialized to $k=\mathbb{Q}$, the paper defines rational Segal categories to be complete dg-Segal spaces over the base field $\mathbb{Q}$, namely objects
\[
F\in \mathrm{Fun}^S(\mathcal{L}_{\mathbb{Q}}^{op},s\mathrm{Set})
\]
satisfying the dg-Segal conditions and the completeness weak equivalence. In this rational case, perfect complexes are bounded complexes of finite-dimensional $\mathbb{Q}$-vector spaces, and the framework specializes cleanly. The main equivalence with dg-categories is stated conditionally: under the hypothesis that DK-equivalences coincide with weak equivalences in the complete dg-Segal model structure, one obtains
\[
\mathrm{Ho}(\mathrm{dg}\text{-}\mathrm{cat})\simeq \mathrm{Ho}(\mathrm{dg}\text{-}\mathrm{Segal}_c)
\]
via the dg-Segal nerve $\mathrm{Sing}$ and its left adjoint [2308.06417].

This usage is not identical to the rational $(\infty,1)$-categorical definition above. The former is dg-enriched and chain-complex based; the latter is formulated through $\mathbb{Q}$-local mapping spaces in simplicial models. The coexistence of these definitions suggests two parallel rationalizations of Segal-type higher categories: one algebraic and one homotopy-theoretic.

## 6. Relations to classical models and one-object cases

The Segal-enriched formalism recovers several established models as special cases. For a monoidal category $M$ and set $X$, an $X$-point $F:P_X\to M$ with Segal maps in $\mathrm{Iso}(M)$ is equivalent to a classical $M$-enriched category. For a bicategory $M$ with $W=2\text{-}\mathrm{iso}(M)$, an $X$-point is equivalent to a category enriched over $M$, or polyad in the sense of Bénabou [1009.3673].

In cartesian targets, the same framework recovers classical Segal categories and Segal $n$-categories. If $M$ has finite products and an appropriate notion of discrete objects, a Segal $n$-category is a simplicial object
\[
A:(\Delta^+)^{op}\to (n-1)\mathrm{SePC}
\]
with Segal maps
\[
A_k\to A_1\times_{A_0}\cdots\times_{A_0}A_1
\]
that are weak equivalences in the $(n-1)$-Segal sense, and this is equivalent to an $X$-point
\[
F:P_X\to (n-1)\mathrm{SePC}
\]
subject to the stated inductive condition [1009.3673].

The one-object case is especially transparent. When $C\simeq 1$, the path-bicategory satisfies
\[
P_1(O,O)\cong (\Delta,+,0)
\]
monoidally. A path-object $F:P_1\to M$ is then equivalent to a colax monoidal functor
\[
(Y,\xi):(\Delta,+,0)\to (M,\otimes,I)
\]
with structure maps
\[
\xi_{m,n}:Y(m+n)\to Y(m)\otimes Y(n),\qquad \xi_0:Y(0)\to I
\]
lying in $W$. Thus a one-object Segal $M_W$-category is exactly an up-to-homotopy monoid in the sense of Leinster [1009.3673].

The rational $(\infty,1)$-categorical model also sits within the standard comparison web of higher-category theories. Rational complete Segal spaces form a cartesian model category, and the rational Segal-category model structure is left-induced from it along the classical adjunction $I\dashv R$. The paper further notes ongoing work toward rational analogues of quasi-categories and simplicial categories, with expected Quillen equivalences among the resulting rational models [2509.22413].

A frequent source of confusion is that “rational Segal category” may refer either to a $\mathbb{Q}$-linear or dg-$\mathbb{Q}$ Segal-enriched category, or to a Segal-category model whose mapping spaces are $\mathbb{Q}$-local spaces. These notions are adjacent but not identical. The former is built from enrichment over $\mathrm{Vect}_{\mathbb{Q}}$ or $\mathrm{Ch}_{\mathbb{Q}}$; the latter is produced by left Bousfield localization at maps detecting rational homotopy groups. Both, however, use the Segal condition to encode composition homotopy-coherently, and both provide rationalized versions of higher categorical structure [1009.3673].

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