Rational Segal Categories Overview

• Rational Segal categories are higher categorical models that blend Segal composition laws with rational, Q-linear, or dg enrichment to encode homotopy-coherent compositions.
• They replace classical fiber products with tensor products through Segal maps and colaxity conditions, enabling robust formulations in both algebraic and homotopical contexts.
• These frameworks bridge classical enriched categories with modern models of rational (∞,1)-categories, establishing equivalences with dg-categories and complete Segal spaces.

Searching arXiv for the cited papers and closely related work on rational Segal categories. {"query": "\"Segal Enriched Categories I\" Bacard (Bacard, 2010) rational Segal categories", "max_results": 5} {"query": "\"A new model of dg-categories\" (Bermejo, 2023) dg-Segal spaces", "max_results": 5} {"query": "\"Models for rational (\infty, 1)-categories\" rational Segal categories (Chatzitheodoridis, 26 Sep 2025)", "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 (Bacard, 2010). In a dg-homotopical setting, complete dg-Segal spaces over $\mathbb{Q}$ provide a model for dg-categories enriched in rational chain complexes (Bermejo, 2023). 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 (Chatzitheodoridis, 26 Sep 2025). 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 $F:P_X\to M$0 is a class of 2-cells called homotopy equivalences. The class $F:P_X\to M$1 is required to contain all invertible 2-cells, satisfy vertical 3-out-of-2, and be stable under horizontal composition. For a small category $F:P_X\to M$2, the associated strict 2-category $F:P_X\to M$3 is the 2-path-category: objects are those of $F:P_X\to M$4, 1-morphisms are finite composable strings in $F:P_X\to M$5, and 2-morphisms are generated by compositions and insertions of identities. Concatenation of chains defines horizontal composition in $F:P_X\to M$6 (Bacard, 2010).

A path-object of $F:P_X\to M$7 is a $F:P_X\to M$8-colax morphism of bicategories

$F:P_X\to M$9

whose colaxity 2-cells, called Segal maps, lie in $\mathrm{Vect}_{\mathbb{Q}}$0. Explicitly, for $\mathrm{Vect}_{\mathbb{Q}}$1 and composable strings $\mathrm{Vect}_{\mathbb{Q}}$2 one has

$\mathrm{Vect}_{\mathbb{Q}}$3

and for each $\mathrm{Vect}_{\mathbb{Q}}$4,

$\mathrm{Vect}_{\mathbb{Q}}$5

When $\mathrm{Vect}_{\mathbb{Q}}$6 is the coarse category on a nonempty set $\mathrm{Vect}_{\mathbb{Q}}$7, such a path-object is called a Segal $\mathrm{Vect}_{\mathbb{Q}}$8-category, or an $\mathrm{Vect}_{\mathbb{Q}}$9-point of $\mathrm{Ch}_{\mathbb{Q}}$0 (Bacard, 2010).

This formalism packages object sets externally and hom-objects internally. The objects are the elements of $\mathrm{Ch}_{\mathbb{Q}}$1, while the hom-object from $\mathrm{Ch}_{\mathbb{Q}}$2 to $\mathrm{Ch}_{\mathbb{Q}}$3 is determined by $\mathrm{Ch}_{\mathbb{Q}}$4. 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 $\mathrm{Ch}_{\mathbb{Q}}$5 (Bacard, 2010).

In cartesian settings, this reproduces the standard simplicial Segal condition. Colax monoidal functors $\mathrm{Ch}_{\mathbb{Q}}$6 correspond to simplicial objects, and the classical Segal maps take the form

$\mathrm{Ch}_{\mathbb{Q}}$7

The enriched formalism replaces those fiber products by tensor products of hom-objects, which is the mechanism that makes linear and dg variants possible (Bacard, 2010).

2. Rational linear and differential graded realizations

In the strict rational linear case, one takes

$\mathrm{Ch}_{\mathbb{Q}}$8

A Segal $\mathrm{Ch}_{\mathbb{Q}}$9-category on an object set $W$0 is then an $W$1-point $W$2 whose Segal maps are isomorphisms. This is equivalent to a classical $W$3-linear enriched category with

$W$4

composition

$W$5

and unit

$W$6

Because the Segal condition is strict here, associativity and unitality hold on the nose (Bacard, 2010).

The dg-rational case replaces vector spaces by chain complexes: $W$7 A Segal DG-category is an $W$8-point $W$9 such that all colaxity maps are quasi-isomorphisms: $\mathbb{Q}$0 and

$\mathbb{Q}$1

Its mapping objects are complexes $\mathbb{Q}$2, and composition is given by a chain map obtained as the composite

$\mathbb{Q}$3

Units are induced by $\mathbb{Q}$4 and $\mathbb{Q}$5 (Bacard, 2010).

In this dg setting, associativity and unitality hold up to $\mathbb{Q}$6, namely in the homotopy category $\mathbb{Q}$7. The differential is compatible with tensor products and composition; for homogeneous $\mathbb{Q}$8,

$\mathbb{Q}$9

and

$(\infty,1)$0

Over a general commutative ring $(\infty,1)$1, the same paper notes that one should use chain homotopy equivalences rather than quasi-isomorphisms, since quasi-isomorphisms are not necessarily stable under $(\infty,1)$2; over $(\infty,1)$3 and, more generally, over fields, quasi-isomorphisms are admissible (Bacard, 2010).

3. Role of the Segal maps and the class $(\infty,1)$4

The class $(\infty,1)$5 is the homotopical control mechanism in Segal enrichment. Requiring the colaxity maps and unit maps to lie in $(\infty,1)$6 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

$(\infty,1)$7

and

$(\infty,1)$8

are the enriched analogues of the usual Segal maps, and their membership in $(\infty,1)$9 is the Segal condition in the noncartesian setting (Bacard, 2010).

This reformulation is significant because it avoids reliance on fiber products. In the classical cartesian case, Segal conditions are stated using iterated pullbacks over $\mathbb{Q}$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 $\mathbb{Q}$1-categories are also organized bicategorically. An $\mathbb{Q}$2-premorphism from $\mathbb{Q}$3 to $\mathbb{Q}$4 is a pair $\mathbb{Q}$5 with a functor $\mathbb{Q}$6 and a transformation of colax morphisms $\mathbb{Q}$7. An $\mathbb{Q}$8-morphism is an $\mathbb{Q}$9-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 $(M,W)$0 (Bacard, 2010).

The same paper provides a canonical reduction process. For any base $(M,W)$1 there exists a bicategory $(M,W)$2 and a homomorphism

$(M,W)$3

that inverts $(M,W)$4 universally. Consequently, a Segal point $(M,W)$5 can be reduced to a strict Segal point $(M,W)$6 in the localized base. Concretely, each hom-category $(M,W)$7 is localized at $(M,W)$8, and composition is induced via the universal property of product localizations (Bacard, 2010).

4. Rational Segal categories as models for rational $(M,W)$9-categories

A later use of the term treats rational Segal categories as Segal-category models for rational $M$0-categories. Here rationality is imposed not by enrichment over $M$1-vector spaces but by requiring mapping spaces to be rational spaces, meaning that for every $M$2 the homotopy group $M$3 is a $M$4-vector space. No constraint is imposed on $M$5 or $M$6. Rationality is encoded model-categorically by localizing simplicial sets at the maps

$M$7

producing a model category whose fibrant objects are precisely the $M$8-local Kan complexes and whose weak equivalences are $M$9-local equivalences (Chatzitheodoridis, 26 Sep 2025).

In this framework, a rational Segal category is a Segal precategory $F:P_X\to M$00 with discrete $F:P_X\to M$01 such that each mapping space $F:P_X\to M$02 is rational, equivalently such that the simplicial space is levelwise rational. The Segal maps

$F:P_X\to M$03

are required to be $F:P_X\to M$04-local weak equivalences (Chatzitheodoridis, 26 Sep 2025).

The corresponding model structure is obtained by left Bousfield localization of Bergner’s $F:P_X\to M$05 at the set

$F:P_X\to M$06

The resulting model category

$F:P_X\to M$07

has monomorphisms as cofibrations, $F:P_X\to M$08-local equivalences as weak equivalences, and Reedy fibrant $F:P_X\to M$09-local Segal categories as fibrant objects (Chatzitheodoridis, 26 Sep 2025).

A rational Dwyer–Kan equivalence between fibrant objects is defined by two conditions: for all objects $F:P_X\to M$10, the induced map

$F:P_X\to M$11

is a $F:P_X\to M$12-local weak homotopy equivalence, and the induced functor on homotopy categories $F:P_X\to M$13 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

$F:P_X\to M$14

Thus rational Segal categories and rational complete Segal spaces are equivalent homotopy-theoretic models for rational $F:P_X\to M$15-categories (Chatzitheodoridis, 26 Sep 2025).

5. dg-Segal spaces over $F:P_X\to M$16

A distinct but adjacent construction models dg-categories by dg-Segal spaces. Fix a commutative ring or field $F:P_X\to M$17. A dg-category $F:P_X\to M$18 over $F:P_X\to M$19 consists of a set of objects, cochain complexes $F:P_X\to M$20, composition maps

$F:P_X\to M$21

and units $F:P_X\to M$22. The paper constructs free dg-categories of finite type and lets $F:P_X\to M$23 be the full simplicial subcategory of the simplicial localization whose objects are cofibrant free dg-categories of finite type (Bermejo, 2023).

A dg-Segal space is a simplicial functor

$F:P_X\to M$24

satisfying three conditions: coproducts are sent to products up to weak equivalence, $F:P_X\to M$25, and for every graph-killing construction $F:P_X\to M$26 the induced square

$F:P_X\to M$27

$F:P_X\to M$28

$F:P_X\to M$29

is a homotopy pullback in simplicial sets. A linearization

$F:P_X\to M$30

produces classical Segal maps

$F:P_X\to M$31

which are weak equivalences for $F:P_X\to M$32 (Bermejo, 2023).

Completeness is encoded by the map

$F:P_X\to M$33

where $F:P_X\to M$34 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 (Bermejo, 2023).

When specialized to $F:P_X\to M$35, the paper defines rational Segal categories to be complete dg-Segal spaces over the base field $F:P_X\to M$36, namely objects

$F:P_X\to M$37

satisfying the dg-Segal conditions and the completeness weak equivalence. In this rational case, perfect complexes are bounded complexes of finite-dimensional $F:P_X\to M$38-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

$F:P_X\to M$39

via the dg-Segal nerve $F:P_X\to M$40 and its left adjoint (Bermejo, 2023).

This usage is not identical to the rational $F:P_X\to M$41-categorical definition above. The former is dg-enriched and chain-complex based; the latter is formulated through $F:P_X\to M$42-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 $F:P_X\to M$43 and set $F:P_X\to M$44, an $F:P_X\to M$45-point $F:P_X\to M$46 with Segal maps in $F:P_X\to M$47 is equivalent to a classical $F:P_X\to M$48-enriched category. For a bicategory $F:P_X\to M$49 with $F:P_X\to M$50, an $F:P_X\to M$51-point is equivalent to a category enriched over $F:P_X\to M$52, or polyad in the sense of Bénabou (Bacard, 2010).

In cartesian targets, the same framework recovers classical Segal categories and Segal $F:P_X\to M$53-categories. If $F:P_X\to M$54 has finite products and an appropriate notion of discrete objects, a Segal $F:P_X\to M$55-category is a simplicial object

$F:P_X\to M$56

with Segal maps

$F:P_X\to M$57

that are weak equivalences in the $F:P_X\to M$58-Segal sense, and this is equivalent to an $F:P_X\to M$59-point

$F:P_X\to M$60

subject to the stated inductive condition (Bacard, 2010).

The one-object case is especially transparent. When $F:P_X\to M$61, the path-bicategory satisfies

$F:P_X\to M$62

monoidally. A path-object $F:P_X\to M$63 is then equivalent to a colax monoidal functor

$F:P_X\to M$64

with structure maps

$F:P_X\to M$65

lying in $F:P_X\to M$66. Thus a one-object Segal $F:P_X\to M$67-category is exactly an up-to-homotopy monoid in the sense of Leinster (Bacard, 2010).

The rational $F:P_X\to M$68-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 $F:P_X\to M$69. The paper further notes ongoing work toward rational analogues of quasi-categories and simplicial categories, with expected Quillen equivalences among the resulting rational models (Chatzitheodoridis, 26 Sep 2025).

A frequent source of confusion is that “rational Segal category” may refer either to a $F:P_X\to M$70-linear or dg-$F:P_X\to M$71 Segal-enriched category, or to a Segal-category model whose mapping spaces are $F:P_X\to M$72-local spaces. These notions are adjacent but not identical. The former is built from enrichment over $F:P_X\to M$73 or $F:P_X\to M$74; 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 (Bacard, 2010).