DIRECTED HOMOTOPY, SECTIONAL INVARIANTS, AND FUNCTORIAL DATABASES
arXiv:2607.23228v1 [math.CT] 25 Jul 2026
ISAAC CARCACÍA-CAMPOS Abstract. A database instance on a small category may be represented as a set-valued functor or, equivalently, as a discrete opfibration. Its sections correspond to globally coherent choices of records. When no global section exists, we measure the failure of global coherence by the minimum number of subcategories on which coherent choices can be made. Regarding natural transformations as directed homotopies, we introduce right and left directed fibrations and relate them to Grothendieck opfibrations and fibrations. We define directed versions of Lusternik-Schnirelmann category and sectional category and establish their invariance and comparison properties. Every functor admits a Grothendieck opfibration model on which directed sectional category is computed by strict local sections, together with a canonical discrete approximation obtained from connected components of comma categories. For functorial databases, we study directed sectional category under decomposition, iteration, and data migration, and characterize initial objects of finite connected acyclic schemas through the existence of global sections of objectwise non-empty databases.
Introduction In the functorial model of databases [10, chapter 3] a database schema is represented by a small category C, while an instance on that schema is a functor X : C −→ Set. The objects of C represent entity types, its morphisms represent functional attributes, X(c) is the set of objects of type c and each morphism X(f ) from a morphism f : c → d represents a morphism between the sets of entities of type X(c) and X(d). 2020 Mathematics Subject Classification. Primary 18A25; Secondary 18A30, 18B40, 55M30, 68P15. Key words and phrases. Directed homotopy, small categories, Grothendieck opfibrations, Lusternik–Schnirelmann category, sectional category, functorial databases, category of elements, data migration. Funding: This research did not receive funding. 1
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For example, the database of a veterinary clinic may contain entity types Dogs and Humans, together with an ownership morphism o : Dogs → Humans. An instance assigns sets of registered dogs and humans, and the function X(o) : X(Dogs) → X(Humans) assigns to every registered dog its registered owner. The category of elements of X determines a way of seeing a database as a category over the base category C Z C X −→ C. πX : where the functor πX has the nice property of being a discrete opfibration. A section of πX as a functor is equivalent to a family of elements xc ∈ X(c), indexed by the objects of C, such that X(f )(xc ) = xd for every morphism f : c → d. Thus, a section represents a globally coherent choice of records across the schema. More general database queries and constraints have also been described categorically in terms of lifting problems [25, 26]. Moreover, for a database X : C → Set, global sections of πX are naturally identified with the elements of limC X. Thus, the existence of a globally coherent selection is an instance of the general problem of deciding whether the limit of a diagram is non-empty. This problem has recently been studied from an algorithmic perspective [1] in the finite (in the sense of having finite diagrams and finite sets), and related existence problems for compatible families arise naturally in combinatorics and constraint satisfaction [14]. This leads to a quantitative refinement of the global section problem: when no globally coherent choice exists, we measure how far the database is from admitting one by determining the minimum number of subcategories covering the schema on which coherent choices can be made. The aim of this article is to introduce categorical invariants that measure this obstruction. The homotopical structure relevant to this problem is intrinsically asymmetric. A natural transformation α : F ⇒ G has a prescribed orientation and need not admit a transformation in the opposite direction. Rather than passing to zigzags of natural transformations, as in the usual homotopy theory of small categories [16], we retain this orientation and regard α as a directed homotopy from F to G.
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This point of view belongs to a broader tradition of using small categories as combinatorial models for homotopical phenomena, which we here develop in an explicitly directed setting. The nerve and classifyingspace constructions associate an ordinary homotopy type with every small category [23, 20] and underlie model structures on small categories [29, 19].Passage to the ordinary homotopy type, however, does not retain the full orientation of the morphisms. Related asymmetric phenomena appear in directed algebraic topology and concurrency theory [9, 12], in the directed homotopy hypothesis [8], and in directed versions of type theory [8, 21]. Our approach is more elementary: we work with ordinary small categories, natural transformations, and Grothendieck fibrations, without introducing higher-categorical or foundational machinery. Directed homotopy for small categories was developed by Grandis [11], who regarded a natural transformation as an oriented homotopy and introduced future and past equivalences of categories. Building on this viewpoint, we consider one-sided notions of domination, homotopy equivalence, and contractibility, closely related to his past and future equivalences. We also use directed homotopies to define directed fibrations and numerical covering invariants. The asymmetry also produces two homotopy lifting properties. Depending on whether the prescribed lift lies over the source or the target of a natural transformation, we obtain right and left directed fibrations. These notions are closely related to Grothendieck opfibrations and fibrations, respectively [13]. We prove that every Grothendieck opfibration is a right directed fibration and, dually, that every Grothendieck fibration is a left directed fibration. We also construct an opfibrational replacement: every functor factors as a right directed homotopy equivalence followed by a Grothendieck opfibration. Using this framework, we introduce two numerical invariants. The first is the directed Lusternik–Schnirelmann category dcat(C), defined by covering C with subcategories whose inclusions admit directed homotopies from constant functors. The second is the directed sectional category dsecat(P ) of a functor P : E → B, defined by requiring local right homotopy sections. These invariants are one-sided analogues of categorical Lusternik–Schnirelmann category and Švarc genus previously studied in the undirected setting [28, 18, 2, 4].
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We establish their basic monotonicity and invariance properties. Directed Lusternik–Schnirelmann category is invariant under right directed homotopy equivalence, while directed sectional category is invariant under compatible directed changes of the total and base categories. In particular, every functor admits a Grothendieck opfibration model with the same directed sectional category, on which local directed homotopy sections can be strictified. We also obtain a canonical discrete lower approximation from the comprehensive factorization [27]. RC For a database X : C → Set, the associated projection πX : X→ C is a discrete opfibration. Hence dsecat(πX ) is computed by strict local sections and measures the minimum normalized number of subcategories needed to cover the schema so that coherent choices of records exist locally. The article is organized as follows. Section 1 introduces directed homotopy, domination, and contractibility. Section 2 develops directed fibrations, compares them with Grothendieck fibrations and opfibrations, and constructs an opfibrational factorization. Section 3 describes discrete opfibrations and their interpretation as functorial databases. Section 4 introduces the morphism covers used in the sequel. Section 5 introduces directed Lusternik–Schnirelmann category and studies its invariance, bounds, and computations for acyclic categories, posets, and monoids. Section 6 develops directed sectional category, establishes its strictification and homotopy-invariance properties, compares its opfibrational and component models, and relates it to directed category and the undirected Švarc genus. Section 7 applies the theory to indecomposable and iterated databases and to data migration, concluding with a characterization of initial objects through global sections. Conventions. Throughout the article, all categories are assumed to be small. Identity morphisms and identity functors are denoted by 1c and 1C , respectively. Sometimes we will avoid writing the composition ◦ to not saturate the reader. A category C is acyclic [15, chapter 10] if every endomorphism is an identity and, for distinct objects c, d ∈ C, the existence of a morphism c → d implies that there is no morphism d → c. 1. Directed Homotopy in Small Categories Homotopies between functors were introduced by Lee [16] and subsequently studied in connection with categorical models of homotopy theory [19]. The usual categorical notion is the equivalence relation generated by natural transformations and hence allows zigzags whose
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arrows may point in either direction. Consequently, the orientation of the individual natural transformations is ultimately forgotten. In this article, we retain this orientation and regard a natural transformation itself as a directed homotopy. This point of view goes back to Grandis [11], who used the walking arrow category I1 = {0 → 1} as a directed interval and identified directed homotopies between functors with natural transformations. Definition 1.1. Let F, G : C → D be functors. A directed homotopy from F to G is a natural transformation α : F ⇒ G. We write F ≤d G when such a natural transformation exists. This definition admits the expected cylinder description. The following description is standard in directed homotopy for categories; see [11, Section 1.5]. Proposition 1.2. Let F, G : C → D be functors. Then F ≤d G if and only if there exists a functor H : C × I1 → D such that H(−, 0) = F and H(−, 1) = G. Proof. A natural transformation α : F ⇒ G determines H by setting H(c, 0) = F (c), H(c, 1) = G(c), and H(1c , s0 ) = αc ; its values on the two copies of C are prescribed by F and G. The naturality of α is precisely the condition required for H to be functorial. Conversely, if such a functor H is given, the morphisms αc = H(1c , s0 ) : F (c) → G(c) form a natural transformation α : F ⇒ G. □ The functor category [C, D] induces a preorder on its objects by declaring F ≤d F ′ ⇐⇒ Nat(F, F ′ ) ̸= ∅. Reflexivity and transitivity follow from identity transformations and vertical composition, respectively. Horizontal composition shows that this preorder is compatible with composition of functors: if F ≤d F ′ , then F ◦ G ≤d F ′ ◦ G and K ◦ F ≤d K ◦ F ′ whenever the corresponding composites are defined. More generally, if F ≤d F ′ and G ≤d G′ , then GF ≤d G′ F ′ whenever these composites are defined. Thus, the hom-categories of Cat become preordered by directed homotopy, and composition is monotone in both variables. In general, this preorder need not be symmetric or antisymmetric. If D is acyclic, however, then [C, D] is also acyclic, and the induced
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preorder is a partial order. We next introduce one-sided versions of domination and homotopy equivalence. Definition 1.3. Let C and D be categories. We say that C is right dominated by D, and write C ⊴R D, if there exist functors F : C → D and G : D → C such that G ◦ F ≤d 1C . Dually, C is left dominated by D, written C ⊴L D, if there exist such functors satisfying 1C ≤d G ◦ F . The categories C and D are right directed homotopy equivalent, written C ≃R D, if C ⊴R D and D⊴R C. Left directed homotopy equivalence, denoted by C ≃L D, is defined dually. Remark 1. Our right and left directed homotopy equivalences are related, respectively, to the past and future equivalences introduced by Grandis [11, Section 2]. More precisely, the orientations of the natural transformations agree, but Grandis additionally imposes coherence identities relating the two transformations. No such coherence conditions are required in the present article. 1.1. Contractibility. Past and future contractibility were studied by Grandis, who related them to initial and terminal objects, respectively [11, Section 2.6]. We now consider the corresponding one-sided notions without imposing the coherence conditions of a past or future equivalence. Let • denote the category with one object and one morphism. A functor C → • is unique, whereas a functor • → C amounts to choosing an object of C. Directed homotopy equivalence with • therefore leads to the following notions. Definition 1.4. An object i ∈ C is homotopically initial if there exists a natural transformation ī ⇒ 1C . Dually, an object t ∈ C is homotopically terminal if there exists a natural transformation 1C ⇒ t̄. Thus, C is right directed homotopy equivalent to • if and only if it has a homotopically initial object. The corresponding left-handed statement is dual. Proposition 1.5. Let C be an acyclic category. Then C ≃R • if and only if C has an initial object. Dually, C ≃L • if and only if C has a terminal object. Proof. We prove the right-handed statement. Suppose that α : ī ⇒ 1C . For every object c, the component αc : i → c provides a morphism from i to c. Since C is acyclic, αi = 1i . If f : i → c is any morphism, naturality with respect to f yields f ◦ αi = αc , and hence f = αc . Therefore i is initial.
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Conversely, if i is initial, the unique morphisms αc : i → c form a natural transformation ī ⇒ 1C . Thus C ≃R •. □ The acyclicity hypothesis is essential. Let M = {1, s} be the monoid defined by s2 = s. Regarded as a one-object category, M admits a natural transformation ∗¯ ⇒ 1M with component s, because m ◦ s = s for every m ∈ M . Nevertheless, its unique object is not initial, since it has two endomorphisms. More generally, a monoid is right directed contractible precisely when it has a right absorbing element; the lefthanded statement is dual. Remark 2. The terminology of right and left agrees with the convention for absorbing elements. If a monoid M is regarded as a one-object category, a natural transformation ∗¯ ⇒ 1M is determined by an element s ∈ M satisfying m ◦ s = s for every m ∈ M , that is, by a right absorbing element. The dual transformation 1M ⇒ ∗¯ corresponds to a left absorbing element. 2. Directed Fibrations in Small Categories The orientation of a directed homotopy distinguishes its two endpoints. A natural transformation F ′ ⇒ G′ cannot in general be reversed, so prescribing a lift over F ′ and prescribing one over G′ lead to different lifting problems. We therefore introduce right and left directed fibrations. We shall compare these notions with Grothendieck fibrations and opfibrations [24, 17, 3]. 2.1. Directed fibrations. For a category C and ε ∈ {0, 1}, let ιCε : C → C × I1 be the functor defined by ιCε (c) = (c, ε). We omit the superscript whenever the category is clear. Definition 2.1. Let P : E → B be a functor. We say that P is a right directed fibration if, for every small category C and every commutative square C
F
ι0
C × I1
E P
H
B,
e : C × I1 → E such that P H e = H and Hι e 0 = F. there exists a functor H Dually, P is a left directed fibration if it has the analogous lifting property with respect to ι1 .
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Remark 3. The right lifting property can equivalently be expressed in terms of natural transformations. Suppose that F ′ , G′ : C → B, that α′ : F ′ ⇒ G′ , and that F : C → E satisfies P F = F ′ . Then there must exist a functor G : C → E and a natural transformation α : F ⇒ G such that P G = G′ and P (αc ) = αc′ for every c ∈ C. For the left lifting property, one starts instead with a lift G of G′ and asks for a lift F of F ′ , together with a natural transformation α : F ⇒ G lifting α′ . 2.2. Grothendieck fibrations and opfibrations. Grothendieck opfibrations lift morphisms covariantly, whereas Grothendieck fibrations provide the corresponding contravariant lifting property. We recall only the opfibrational definitions; the fibrational versions are dual. See [24, 17, 3] for further details. Definition 2.2. Let P : E → B be a functor. A morphism φ : e1 → e2 in E is P -opcartesian if, for every morphism β : e1 → e in E and every morphism α : P (e2 ) → P (e) in B satisfying α ◦ P (φ) = P (β), there exists a unique morphism α : e2 → e such that α ◦ φ = β and P (α) = α. Equivalently, every diagram of the following form admits a unique dashed factorization. e1 φ
e2
β
α
e
P e1 Pφ
Pβ
Pe
α
P e2
Definition 2.3. A functor P : E → B is a Grothendieck opfibration if, for every morphism φ : b1 → b2 in B and every e1 ∈ E with P (e1 ) = b1 , there exists a P -opcartesian morphism φ : e1 → e2 satisfying P (φ) = φ. Remark 4. The dual notions of P -cartesian morphism and Grothendieck fibration are obtained by reversing all arrows. Equivalently, P : E → B is a Grothendieck fibration if and only if P op : E op → B op is a Grothendieck opfibration. We shall state most results in their opfibrational form; the corresponding fibrational statements follow by duality. See [24, 13, 17, 3] for further background. The Grothendieck construction gives a systematic method for producing opfibrations from covariantly indexed categories. Definition 2.4. Let P : RB → Cat be a functor. Its Grothendieck B construction, denoted by P , is the category whose objects are pairs (b, x), where b ∈ B and x ∈ P (b). A morphism (φ, f ) : (b, x) −→ (b′ , y)
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consists of a morphism φ : b → b′ in B and a morphism f : P (φ)(x) → y in P (b′ ). If (ψ, g) : (b′ , y) → (b′′ , z), their composite is (ψ, g) ◦ (φ, f ) = ψ ◦ φ, g ◦ P (ψ)(f ) . RB The projection πP : P → B is defined by πP (b, x) = b and πP (φ, f ) = φ. RB Proposition 2.5. Let P : B → Cat be a functor. Then πP : P →B op is a split Grothendieck opfibration. Dually, a functor P : B → Cat determines a split Grothendieck fibration over B. Proof. Let φ : b1 → b2 be a morphism in B, and let (b1 , x) lie over b1 . Consider (φ, 1P (φ)(x) ) : (b1 , x) −→ (b2 , P (φ)(x)). We show that this morphism is opcartesian. Suppose that (β, g) : (b1 , x) → (b, y) and that α : b2 → b satisfy α ◦ φ = β. Then g : P (β)(x) = P (α)(P (φ)(x)) −→ y, so (α, g) is a morphism from (b2 , P (φ)(x)) to (b, y). It is the unique morphism over α whose composite with (φ, 1P (φ)(x) ) is (β, g). Hence the latter is opcartesian. These chosen lifts preserve identities and composition, so πP is split. □ Proposition 2.6. Every Grothendieck opfibration is a right directed fibration. Dually, every Grothendieck fibration is a left directed fibration. Conversely, a right directed fibration is a Grothendieck opfibration if its liftings can always be chosen so that every component of the lifted natural transformation is opcartesian. Proof. Let P : E → B be a Grothendieck opfibration. Consider functors F ′ , G′ : C → B, a natural transformation α′ : F ′ ⇒ G′ , and a lift F : C → E of F ′ . Choose, for every c ∈ C, an opcartesian lift αc : F (c) → G(c) of αc′ : F ′ (c) → G′ (c). For a morphism f : c → c′ , naturality of α′ gives G′ (f ) ◦ αc′ = αc′ ′ ◦ F ′ (f ). Since αc is opcartesian, there is a unique morphism G(f ) : G(c) → G(c′ ) over G′ (f ) such that G(f ) ◦ αc = αc′ ◦ F (f ).
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Uniqueness shows that G preserves identities and composition. Thus G : C → E is a functor and the family (αc )c∈C defines a natural transformation α : F ⇒ G lifting α′ . Hence P is a right directed fibration. For the converse, apply the lifting property with C = •. A morphism in B, together with an object of E over its source, determines a natural transformation indexed by •. By hypothesis, its lift may be chosen opcartesian, which is precisely the lifting condition for a Grothendieck opfibration. □ Remark 5. The proof uses a simultaneous choice of opcartesian lifts. This is automatic for cloven opfibrations and canonical for split opfibrations. For arbitrary small opfibrations, it uses the axiom of choice. 2.3. Factorization by opfibrations. Comma categories play the role of directed homotopy pullbacks in the framework of Grandis, where they are also used to factor adjunctions and construct directed models [11, Sections 1.6 and 4.4]. We use a related comma-category construction to obtain an opfibrational factorization of an arbitrary functor. More precisely, we will show that every functor factors as a right directed homotopy equivalence followed by a Grothendieck opfibration. Proposition 2.7. Every functor F : C → D factors as F
F
1 2 C −→ EF −→ D,
where F1 is a right directed homotopy equivalence and F2 is a Grothendieck opfibration. Proof. Let EF = F ↓ 1D . We write its objects as morphisms h : F (c) → d, and its morphisms as pairs (g, u) : h −→ h′ satisfying uh = h′ F (g). Define F1 (c) = 1F (c) , F1 (g) = (g, F (g)), and let R : EF → C send h : F (c) → d to c and (g, u) to g. Then RF1 = 1C , while the morphisms (1c , h) : F1 R(h) −→ h form a natural transformation F1 R ⇒ 1EF . Hence F1 is a right directed homotopy equivalence. Define F2 (h : F (c) → d) = d and F2 (g, u) = u. Then F2 F1 = F . Given u : d → d′ and an object h : F (c) → d, the morphism (1c , u) : h −→ uh
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lies over u. If (g, v) : h → h′ lies over v = wu, then h′ F (g) = vh = wuh, so there is a unique morphism (g, w) : uh −→ h′ over w whose composite with (1c , u) is (g, v). Thus (1c , u) is opcartesian, and F2 is a Grothendieck opfibration. □ Remark 6. The opfibration F2 is the Grothendieck construction of the functor F ↓ (−) : D −→ Cat, d 7−→ F ↓ d. ′ A morphism u : d → d induces a functor F ↓ d → F ↓ d′ by postcomposition, h 7→ u ◦ h. Remark 7. The construction admits a dual version. Using the comma category 1D ↓ F , every functor F : C → D factors as a left directed homotopy equivalence followed by a Grothendieck fibration. We emphasize the opfibrational factorization because a functor X : C → Set, interpreted as a database, determines a discrete opfibration through its Grothendieck construction. 3. Discrete Opfibrations and Databases A Grothendieck opfibration may be regarded as a family of categories indexed by the objects of its base. When the fibres are discrete, this family is set-valued and the lifting property becomes an existence-anduniqueness condition. Definition 3.1. A functor P : E → B is a discrete opfibration if, for every morphism f : b → b′ in B and every e ∈ E with P (e) = b, there exists a unique morphism f : e → e′ satisfying P (f ) = f . Thus, both the lift f and its target are uniquely determined by f and e. Proposition 3.2. Every morphism of a discrete opfibration P : E → B is P -opcartesian. In particular, every discrete opfibration is a Grothendieck opfibration. Proof. Let φ : e1 → e2 , and suppose that β : e1 → e and α : P (e2 ) → P (e) satisfy αP (φ) = P (β). Let α : e2 → e′ be the unique lift of α with source e2 . The morphisms αφ and β have the same source and lie over the same morphism. Uniqueness of lifts gives αφ = β and e′ = e. The required factorization is therefore unique, so φ is opcartesian. □ Conversely, an opfibration is discrete precisely when its fibres are discrete.
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Proposition 3.3. Let P : E → B be a Grothendieck opfibration. If every fibre Eb is discrete, then P is a discrete opfibration. Proof. Let f : e → e′ be an opcartesian lift of f : b → b′ . If γ : e → e′′ is another lift, the opcartesian property gives a unique morphism α : e′ → e′′ over 1b′ such that αf = γ. Since α belongs to the discrete fibre Eb′ , it is an identity. Hence e′ = e′′ and γ = f . □ 3.1. The Category of Elements. Let X : B → Set. Its category RB of elements X has objects (b, x), with x ∈ X(b), and a morphism ′ ′ (b, x) → (b , x ) is a morphism f : b → b′ satisfying X(f )(x) = x′ . The projection Z B πX : X −→ B sends (b, x) to b and every morphism over f to f . The Grothendieck construction induces the standard equivalence [B, Set] ≃ DOpFib(B) between set-valued functors and discrete opfibrations over B; see [17, 25]. Under this equivalence, a discrete opfibration P : E → B corresponds to the functor b 7−→ Obj(Eb ), whose action on f : b → b′ sends e ∈ Eb to the target f! e of the unique lift of f with source e. 3.2. Functorial Databases. In the functorial model of databases, a schema is a small category B and an instance on that schema is a functor X : B → Set [10, 25, 26]. An object b ∈ B represents an entity type, X(b) is its set of records, and a morphism f : b → b′ represents a functional attribute or foreign key. The function X(f ) : X(b) → X(b′ ) assigns to each record its corresponding value under that attribute. Example 3.4. Consider the schema BDCH generated by Dogs oD
Cats
oC
Humans.
The morphisms oD and oC represent ownership. An instance assigns sets of registered dogs, cats, and humans, together with functions assigning to every registered animal its unique registered owner.
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For example, let X(Dogs) = {d1 , d2 , d3 }, X(Cats) = {c1 , c2 }, X(Humans) = {h1 , h2 , h3 }, with X(oD )(d1 ) = X(oD )(d2 ) = h1 , X(oD )(d3 ) = X(oC )(c1 ) = h2 , X(oC )(c2 ) = h3 . The corresponding category of elements is represented by (Dog, d1 ) (Dog, d2 )
(Human, h1 )
(Dog, d3 )
(Human, h2 )
(Cat, c1 ) (Cat, c2 ) (Human, h3 ). The projection πX sends every object to its entity type and every morphism to the corresponding ownership morphism. 3.3. Sections and Globally Coherent Data. Proposition 3.5. Let X : B → Set. A section of πX : equivalent to a family (xb )b∈B , with xb ∈ X(b), satisfying
RB
X → B is
X(f )(xb ) = xb′ for every morphism f : b → b′ . Proof. If s is a section, write s(b) = (b, xb ). For f : b → b′ , the morphism s(f ) exists in the category of elements precisely when X(f )(xb ) = xb′ . Conversely, a family satisfying these equations defines a section by s(b) = (b, xb ) and s(f ) = f . □ Thus, a section is a globally coherent choice of one record of every entity type: the selected records agree with every attribute and foreign-key map. Equivalently, a section determines an element of the
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limit of X. When such a global choice does not exist, one may ask whether coherent choices exist after restricting the schema to suitable subcategories. This will be the key idea behind Section 6. Example 3.6. Let M be a monoid regarded as a one-object category. A functor X : M → Set is a left action of M on X(∗). RM The category of elements X is the corresponding action category: its objects are the elements x ∈ X(∗), and a morphism labelled by m ∈ M from x to y exists R M precisely when m · x = y. A section of πX : X → M is therefore an element x ∈ X(∗) satisfying m·x = x for every m ∈ M . Thus global sections are precisely the common fixed points of the action. 3.4. The comprehensive factorization. The comprehensive factorization of Street and Walters associates a canonical database with every functor [27]. It will later provide a discrete lower approximation to directed sectional category. Let F : C → D. Define its component database KF : D → Set by KF (d) = π0 (F ↓ d), where π0 denotes connected components with respect to unoriented zigzags. A morphism u : d → d′ acts by postcomposition: KF (u)[c, α] = [c, uα]. Let
Z D qF :
KF −→ D
be the associated discrete opfibration. There is a canonical functor Z D iF : C −→ KF , iF (c) = F (c), [(c, 1F (c) )] , and F = qF iF . Proposition 3.7 (Street–Walters). Every functor F : C → D admits a functorial factorization Z D qF iF C −→ KF −→ D, where iF is initial and qF is a discrete opfibration. Proof. The functor qF is a discrete opfibration because it is a categoryRD of-elements projection. For (d, ξ) ∈ KF , the comma category iF ↓ (d, ξ) is naturally isomorphic to the full subcategory of F ↓ d determined by the component ξ. It is therefore non-empty and connected, so iF is initial. □
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For Grothendieck opfibrations, the comprehensive factorization amounts to taking connected components fibrewise. Proposition 3.8. Let X : C → Cat, and let Z C PX : X −→ C be its Grothendieck opfibration. Then KP ∼ (π0 X)(c) = π0 (X(c)), = π0 X, X
and the comprehensive factorization of PX is naturally isomorphic to Z C Z C π π0 X κX X −→ π0 X −−− → C, where κX (c, x) = (c, [x]). Proof. For every d ∈ C, an object of PX ↓ d has the form (c, x), f : c → d . The opcartesian lift of f connects this object to (d, X(f )(x)), 1d . Thus every component of PX ↓ d contains an object of the fibre X(d), and two such objects belong to the same component precisely when they are connected in X(d). Hence π0 (PX ↓ d) ∼ = π0 (X(d)), naturally in d, and therefore KPX ∼ = π0 X.
□
Remark 8. Thus, the comprehensive factorization of a Grothendieck opfibration replaces each categorical fibre by its set of connected components. In particular, if PX is a discrete opfibration, then every X(d) is discrete, so π0 X ∼ = X and ππ X ∼ = PX 0
over C. 4. Covers of Small Categories The invariants introduced below are defined by decomposing a category into subcategories on which a prescribed homotopical or sectional condition holds. We first specify the notion of cover used throughout the article. Definition 4.1. A morphism cover of a category C is a family of subcategories {Ui }i∈I such that every morphism of C belongs to at least one Ui .
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Since every identity morphism 1c belongs to some member of the cover, each object c ∈ C also belongs to at least one Ui . Throughout the remainder of the article, cover will always mean a morphism cover in the sense of Definition 4.1. Remark 9. Morphism covers are weaker than the geometric covers commonly used in categorical versions of Lusternik–Schnirelmann category, sectional category, and homotopic distance [28, 18, 2, 4]. A geometric cover requires every finite composable sequence of morphisms to be contained in one member of the cover. This stronger requirement is natural when the categorical invariant is intended to model an invariant of the nerve or classifying space, since the simplices of the nerve are precisely composable sequences of morphisms. Our purpose is different: we retain the directed categorical structure itself and use subcategories as local domains on which directed deformations or coherent selections may be defined. In particular, for a database X : C → Set, the members of a cover represent regions of the schema on which compatible choices of records can be made. Morphism covers are also preserved by inverse images, a property used repeatedly below. Lemma 4.2. Let F : C → D be a functor. If {Ui }i∈I is a cover of D, then {F −1 (Ui )}i∈I is a cover of C. Proof. For every morphism f of C, the morphism F (f ) belongs to some Ui . Hence f ∈ F −1 (Ui ). □ We use the normalized convention for all sectional invariants: an invariant equal to n corresponds to a cover by n + 1 subcategories. 5. Directed Lusternik–Schnirelmann Category The Lusternik–Schnirelmann category is a classical invariant measuring the minimum number of open subsets that are null-homotopic in the ambient space; see [5]. A directed Lusternik–Schnirelmann category for directed topological spaces was recently introduced by Datta, Daundkar, and Sarkar [6]. Categorical analogues for the usual, nondirected, LS-category have been defined using zigzags of natural transformations [28, 4]. The invariant defined below is of a different nature: it is formulated directly for small categories, using morphism covers and oriented natural transformations. Definition 5.1. The directed Lusternik–Schnirelmann category of a small category C, denoted by dcat(C), is the least integer n ≥ 0 for
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which C admits a cover {U0 , . . . , Un } such that, for every i, there exist an object ci ∈ C and a natural transformation αi : ci ⇒ ιi , where ci : Ui → C is the constant functor with value ci and ιi : Ui ,→ C is the inclusion. Such a subcategory is called right categorical, and a cover by right categorical subcategories is called a right categorical cover. If no finite right categorical cover exists, we set dcat(C) = ∞. Thus, dcat(C) measures the minimum number of local pieces on which a constant functor can be deformed towards the corresponding inclusion and the value zero recovers directed contractibility. Proposition 5.2. Let C be a non-empty small category. Then dcat(C) = 0 if and only if C has a homotopically initial object. If, moreover, C is acyclic, these conditions are equivalent to the existence of an initial object. Proof. The equality dcat(C) = 0 means precisely that the cover {C} is right categorical, that is, that there exist an object i ∈ C and a natural transformation i ⇒ 1C . Thus i is homotopically initial. The statement for acyclic categories follows from Proposition 1.5. □ 5.1. Domination and Directed Homotopy Equivalence. Proposition 5.3. Let C and D be small categories. If C ⊴R D, then dcat(C) ≤ dcat(D). Consequently, if C ≃R D, then dcat(C) = dcat(D). Proof. Choose F : C → D, G : D → C, and γ : GF ⇒ 1C . The assertion is immediate if dcat(D) = ∞. Otherwise, let {U0 , . . . , Un } be a minimal right categorical cover, with di ≤d ιi . Set Vi = F −1 (Ui ), and denote by ι′i : Vi ,→ C the inclusion. Applying G, restricting along F |Vi , and then using γ, gives G(di ) ≤d GF ι′i ≤d ι′i . Thus every Vi is right categorical. Since the Vi cover C, we obtain dcat(C) ≤ dcat(D). The equality for right directed homotopy equivalent categories follows by applying the inequality in both directions. □
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5.2. Bounds for Finite Acyclic Categories. Definition 5.4. Let C be an acyclic category. An object s ∈ C is a source if HomC (x, s) = ∅ for every x ̸= s. Thus, a source receives no non-identity morphisms. Remark 10. Every finite non-empty acyclic category has a source. Indeed, starting at any object and repeatedly following a non-identity incoming morphism must terminate: acyclicity prevents repetition and finiteness prevents an infinite sequence. Proposition 5.5. Let C be a finite non-empty acyclic category. Then dcat(C) ≥ | Src(C)| − 1. Proof. Suppose that {U0 , . . . , Un } is a right categorical cover, with transformations αi : ci ⇒ ιi . If s is a source, then s ∈ Ui for some i, since 1s belongs to the cover. The component αsi : ci → s forces ci = s. Thus every source occurs among c0 , . . . , cn , so | Src(C)| ≤ n + 1. Taking a minimal cover gives the result. □ For finite posets, this lower bound is attained. Proposition 5.6. Let P be a finite non-empty poset, regarded as a category, and let r be the number of its minimal elements. Then dcat(P ) = r − 1. Proof. The minimal elements of P are its sources, so Proposition 5.5 gives dcat(P ) ≥ r − 1. Let s1 , . . . , sr be the minimal elements, and let Uj be the full subcategory on the principal upper set ↑ sj = {x ∈ P | sj ≤ x}. Every element of P lies above a minimal element. Moreover, if x ≤ y and sj ≤ x, then sj ≤ y. Hence the Uj cover all morphisms of P . For every x ∈ Uj , the unique morphism sj → x defines the component of a natural transformation sj ⇒ ιj . Thus the Uj form a right categorical cover and dcat(P ) ≤ r − 1. □ We next give an upper bound in terms of maximal chains. An nchain in C is a sequence f1
f2
fn
c0 − → c1 − → · · · −→ cn . Equivalently, it is an n-simplex of the nerve N C [22]. It is nondegenerate if none of the fi is an identity; objects are regarded as
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non-degenerate 0-chains. A non-degenerate chain is maximal if it is not a proper face of another non-degenerate chain. We denote the set of maximal chains by MaxChains(C). Proposition 5.7. Let C be a finite non-empty acyclic category. Then dcat(C) ≤ | MaxChains(C)| − 1. Proof. Every non-degenerate chain lies in a maximal one. For a maximal chain f1 f2 fn f¯ = c0 − → c1 − → · · · −→ cn , let Uf¯ be the subcategory generated by its morphisms. For every ci ∈ Uf¯, the composite fi ◦ · · · ◦ f1 : c0 → ci defines the component of a natural transformation c0 ⇒ ιf¯. Thus Uf¯ is right categorical. Every non-identity morphism is a non-degenerate 1-chain and hence lies in a maximal chain. Every object, regarded as a 0-chain, also lies in a maximal chain. Therefore the subcategories Uf¯ form a right categorical cover, and dcat(C) + 1 ≤ | MaxChains(C)|. □ For an acyclic category C, its subdivision sub(C) is the poset of non-degenerate chains ordered by the face relation [7]. In the undirected setting, subdivision does not increase categorical LS-category: ccat(sub(C)) ≤ ccat(C) [28, Corollary 3.4 and Proposition 2.19]. The directed invariant satisfies the following inequality in the opposite direction. Corollary 5.8. Let C be a finite non-empty acyclic category. Then dcat(C) ≤ dcat sub(C)op . Proof. The sources of sub(C)op are the maximal chains of C. By Proposition 5.6, dcat sub(C)op = | MaxChains(C)| − 1. Now apply Proposition 5.7.
□
5.3. Comparison with Undirected LS-Category. To isolate the effect of orienting the homotopies while retaining the same covers, we introduce the following undirected invariant. Definition 5.9. The strong normalized categorical Lusternik–Schnirelmann category of a small category C, denoted by ccat(C), is the least integer
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n ≥ 0 for which C admits a cover {U0 , . . . , Un } such that every inclusion ιi : Ui ,→ C is strongly homotopic to a constant functor. If no finite cover exists, we set ccat(C) = ∞. Remark 11. Although Definition 5.9 uses the usual undirected homotopy relation, its covers are the morphism covers of Definition 4.1. Thus, ccat differs from Tanaka’s categorical LS-category, which uses geometric covers [28]. Proposition 5.10. For every small category C, ccat(C) ≤ dcat(C). Proof. A natural transformation ci ⇒ ιi is a zigzag of length one. Hence every right categorical cover is also strongly categorical. □ 5.4. The Case of Monoids. Let M be a monoid regarded as a oneobject category. Its subcategories are precisely its submonoids, and a cover of M is therefore a family of submonoids whose union is M . Proposition 5.11. Let N ⊆ M be a submonoid. Then N is right categorical in M if and only if there exists s ∈ M such that ns = s for every n ∈ N . Proof. There is a unique constant functor ∗ : N → M . A natural transformation α : ∗ ⇒ ιN is determined by its component s = α∗ ∈ M . Naturality with respect to n ∈ N is the commutativity of ∗
s
n
1
∗
∗
s
∗,
which is equivalent to ns = s.
□
Remark 12. The element s need not belong to N . It is an element of the ambient monoid fixed under left multiplication by N . If s ∈ N , then s is a right absorbing element of N . Proposition 5.12. Let M be a monoid. Then dcat(M ) ≤ r − 1 if and only if there exist submonoids N1 , . . . , Nr ⊆ M and elements s1 , . . . , sr ∈ M such that M = N1 ∪ · · · ∪ Nr
and
nsi = si
for every n ∈ Ni .
Proof. This follows directly from Proposition 5.11: the Ni form a right categorical cover precisely under the stated conditions. □ Corollary 5.13. A monoid M satisfies dcat(M ) = 0 if and only if it has a right absorbing element. Proof. Apply Proposition 5.12 with r = 1.
□
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Proposition 5.14. Let G be a group regarded as a one-object category. Then ( 0, if G = {1}, dcat(G) = ∞, if G ̸= {1}. Proof. The trivial group is the terminal category. If N ⊆ G is right categorical, there exists s ∈ G such that ns = s for every n ∈ N . Cancellation gives n = 1, so N = {1}. Hence no family of right categorical submonoids can cover a nontrivial group. □ 5.5. Examples. The following examples exhibit the asymmetry of directed LS-category. Example 5.15. Let I2 be the category generated by a1 f
a2
b.
g
The category I2 has two sources and two maximal chains. Propositions 5.5 and 5.7 therefore give dcat(I2 ) = 1. op By contrast, b is initial in Iop 2 , so dcat(I2 ) = 0. Thus directed LScategory is not invariant under passage to the opposite category. Example 5.16. For n ≥ 1, let In be the category generated by a1 .. . ai .. .
f1 fi
b
fn
an The category has n sources and n maximal chains, so dcat(In ) = n − 1. On the other hand, b is terminal. Hence 1In ⇒ b, so ccat(In ) = 0. Therefore dcat(In ) − ccat(In ) = n − 1, and the difference between the directed and undirected invariants is unbounded.
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Example 5.17. Let P be the category generated by f1
x
g1
y
z g2
f2
subject to gj ◦ fi = gl ◦ fk (i, j, k, l ∈ {1, 2}). Thus all four composites from x to z coincide. The object x is the unique source but is not initial, since f1 ̸= f2 . Hence dcat(P) ̸= 0. For i ∈ {1, 2}, let Ui be the subcategory generated by g1
x
fi
y
z. g2
The subcategories U1 and U2 cover P. Moreover, x is initial in each Ui : the unique morphism x → y is fi , and the defining relations imply g1 fi = g2 fi , giving a unique morphism x → z. Thus both subcategories are right categorical, and dcat(P) = 1. 6. Directed Sectional Category Sectional category measures the local existence of sections of a functor. We first recall the strict version introduced in [2]. Definition 6.1. The sectional category of a functor P : E → B, denoted by secat(P ), is the least integer n ≥ 0 for which B admits a cover {U0 , . . . , Un } and, for every i, a functor si : Ui → E satisfying P ◦ si = ιi , where ιi : Ui ,→ B is the inclusion. The functor si is called a local section of P over Ui . If no such finite cover exists, we set secat(P ) = ∞. For an arbitrary functor, strict local sections may be too rigid. Replacing them by homotopy sections gives the categorical analogue of the Švarc genus considered in [2, 4]. We use right directed homotopies, since a database determines a discrete opfibration and Grothendieck opfibrations satisfy the right directed homotopy lifting property. The left-handed theory is dual. Definition 6.2. The directed sectional category of a functor P : E → B, denoted by dsecat(P ), is the least integer n ≥ 0 for which B admits a
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cover {U0 , . . . , Un } and, for every i, a functor si : Ui → E together with a natural transformation αi : P ◦ si ⇒ ιi . If no such finite cover exists, we set dsecat(P ) = ∞. The pair (si , αi ) is called a local right homotopy section. Equivalently, its defining condition is P ◦ si ≤d ιi . 6.1. Composition. Directed sectional category is monotone under composition: every local right homotopy section of a composite P F induces one of P . Proposition 6.3. Let P : E → B and F : C → E. Then dsecat(P ) ≤ dsecat(P F ). Proof. If si : Ui → C and αi : P F si ⇒ ιi are local right homotopy sections of P F , then F si : Ui → E and the same transformation αi are local right homotopy sections of P . □ 6.2. Opfibrations and Strictification. For Grothendieck opfibrations, local right homotopy sections can be strictified. Proposition 6.4. For every functor P : E → B, dsecat(P ) ≤ secat(P ). If P is a Grothendieck opfibration, then dsecat(P ) = secat(P ). Proof. Every strict local section is a local right homotopy section, using the identity transformation. Hence dsecat(P ) ≤ secat(P ). Suppose that P is a Grothendieck opfibration and that s : U → E is a local right homotopy section, with α : P s ⇒ ι. Let H : U × I1 → B be the homotopy corresponding to α. We have a commutative square U
s
ι0
U × I1
E P
H
B.
By Proposition 2.6, P is a right directed fibration. Hence H admits e : U × I1 → E satisfying Hι e 0 = s. The functor s = Hι e 1 satisfies a lift H e 1 = Hι1 = ι, P s = P Hι and is therefore a strict local section. Thus every local right homotopy section of P can be strictified. □
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Corollary 6.5. A Grothendieck opfibration P : E → B satisfies dsecat(P ) = 0 if and only if it admits a global section. Proof. This follows immediately from Proposition 6.4.
□
6.3. Monotonicity and Homotopy Invariance. Directed sectional category is monotone with respect to the directed homotopy preorder. Indeed, a directed deformation from P to P ′ allows every local right homotopy section of P ′ to be regarded as one of P . We first record this elementary observation and then use it to obtain an invariance result for functors whose total and base categories are related by compatible directed comparison data. Proposition 6.6. Let P, P ′ : E → B be functors. If P ≤d P ′ , then dsecat(P ) ≤ dsecat(P ′ ). Proof. Let η : P ⇒ P ′ , and suppose that si : Ui → E is a local right homotopy section of P ′ . Then P si ≤d P ′ si ≤d ιi . By transitivity, si is a local right homotopy section of P . Thus every cover witnessing dsecat(P ′ ) ≤ n also witnesses dsecat(P ) ≤ n. □ Proposition 6.6 compares functors with the same domain and codomain. We now allow both categories to vary. To transfer local sections, we require functors between the total categories and between the bases, together with directed homotopies expressing that the resulting squares commute up to the prescribed orientation. Mutual right dominations of the base categories then provide the comparison in both directions. Theorem 6.7. Consider a diagram F1
E′
E P
G1
P′
F2
B′
B G2
together with natural transformations λ : P ′ F1 ⇒ F2 P,
µ : P G1 ⇒ G2 P ′ .
Suppose that there are also natural transformations α : G2 F2 ⇒ 1B ,
β : F2 G2 ⇒ 1B′ .
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Then dsecat(P ) = dsecat(P ′ ). Proof. We first show that dsecat(P ) ≤ dsecat(P ′ ). Let {U0 , . . . , Un } be a cover of B ′ admitting local right homotopy sections s′i : Ui −→ E ′ ,
P ′ s′i ≤d ι′i .
Set Vi = F2−1 (Ui ), let F2,i : Vi → Ui be the restriction of F2 , and define si = G1 s′i F2,i : Vi −→ E. The given transformations and the transitivity of ≤d yield P si = P G1 s′i F2,i ≤d G2 P ′ s′i F2,i ≤d G2 ι′i F2,i = G2 F2 ιi ≤d ιi . Thus si is a local right homotopy section of P . Since the Vi cover B, we obtain dsecat(P ) ≤ dsecat(P ′ ). For the reverse inequality, start with a cover of B, take its inverse images under G2 , and use F1 , λ, and β. □ Remark 13. The strictly commutative case is recovered by taking λ and µ to be identity transformations. The principal application is that the opfibrational replacement of an arbitrary functor preserves directed sectional category. Corollary 6.8. Let P : E → B, and consider the factorization P
E P1
B P2
EP of Proposition 2.7. Then dsecat(P ) = dsecat(P2 ) = secat(P2 ). Proof. Let R : EP → E and η : P1 R ⇒ 1EP be those constructed in Proposition 2.7. Apply Theorem 6.7 with F1 = P1 ,
G1 = R,
F2 = G2 = 1B .
The required compatibility transformations are the identity P2 P1 = P and the transformation P R = P2 P1 R ⇒ P2 obtained by applying P2 to η. Hence dsecat(P ) = dsecat(P2 ). Since P2 is a Grothendieck opfibration, Proposition 6.4 gives the second equality. □
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6.4. Component sectional category. The opfibrational replacement retains the full directed sectional obstruction. We now compare it with the discrete model provided by the comprehensive factorization. Let P : E → B be a functor. Recall from Subsection 3.4 that its comprehensive factorization is P
E
B qP
iP
Z B KP where KP (b) = π0 (P ↓ b), the functor iP is initial, and qP is a discrete opfibration. The database KP retains only the connected components of the comma categories P ↓ b. It therefore provides a discrete approximation to the local section problem for P . Definition 6.9. The component sectional category of a functor P : E → B is csecat(P ) := dsecat(qP ). Since qP is a discrete opfibration, its directed sectional category agrees with its strict sectional category. Thus csecat(P ) measures the minimum normalized number of subcategories needed to cover B so that compatible connected components of the comma categories P ↓ b can be selected locally. Proposition 6.10. For every functor P : E → B, csecat(P ) ≤ dsecat(P ). Proof. The comprehensive factorization satisfies P = qP iP . Hence Proposition 6.3 gives dsecat(qP ) ≤ dsecat(qP iP ) = dsecat(P ). □ The preceding inequality admits a description purely in terms of limits. Proposition 6.11. Let P : E → B be a functor. Then csecat(P ) ≤ n if and only if B admits a cover {U0 , . . . , Un } such that lim KP |Ui ̸= ∅ Ui
for every i.
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Proof. Since qP is the category-of-elements projection of KP , a section of qP over Ui is equivalent to an element of lim KP |Ui . Ui
Moreover, qP is a discrete opfibration, so local right homotopy sections can be strictified by Proposition 6.4. The result follows from the definition of csecat(P ). □ Suppose now that P is the Grothendieck opfibration associated with a functor X : B → Cat. As shown in Proposition 3.8, its component database is naturally isomorphic to π0 X : B −→ Set,
b 7−→ π0 (X(b)).
Corollary 6.12. Let X : B → Cat, and let Z B PX : X −→ B be its Grothendieck opfibration. Then csecat(PX ) = dsecat(ππ0 X ) ≤ dsecat(PX ). Proof. The discrete opfibration in the comprehensive factorization of PX is naturally isomorphic to Z B ππ 0 X : π0 X −→ B. The equality follows from Definition 6.9, and the inequality from Proposition 6.10. □ Combining the opfibrational and comprehensive factorizations gives the following comparison. Theorem 6.13. Let P : E → B be a functor. Let P
P
1 2 E −→ EP −→ B
be its directed opfibrational factorization, and let Z B qP iP E −→ KP −→ B be its comprehensive factorization. Then csecat(P ) = dsecat(qP ) ≤ dsecat(P ) = dsecat(P2 ). Proof. The inequality is Proposition 6.10. The final equality follows from Corollary 6.8. □
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Thus every functor admits two canonical opfibrational models. The Grothendieck opfibration P2 retains the full directed sectional obstruction, whereas the discrete opfibration qP records only the obstruction visible at the level of connected components. 6.5. Relations with Directed LS-Category. Proposition 6.14. Let P : E → B be surjective on objects. Then dsecat(P ) ≤ dcat(B). Proof. Let {U0 , . . . , Un } be a right categorical cover of B, with transformations αi : bi ⇒ ιi . Choose ei ∈ E with P (ei ) = bi , and let si : Ui → E be constant at ei . Then P si is constant at bi , so αi : P si ⇒ ιi . Thus the same cover witnesses dsecat(P ) ≤ dcat(B). □ For finite posets, it is enough that the fibres over the sources be non-empty. Corollary 6.15. Let P : E → Q, where Q is a finite non-empty poset. If P −1 (s) ̸= ∅ for every source s of Q, then dsecat(P ) ≤ dcat(Q). Proof. Let s1 , . . . , sr be the sources of Q. The principal upper sets Ui =↑ si form a right categorical cover. Choose ei ∈ P −1 (si ). The constant functor at ei is a local right homotopy section over Ui , since si is initial in Ui . Hence dsecat(P ) ≤ r − 1 = dcat(Q). □ 6.6. Relations with the Švarc Genus. To isolate the effect of orienting the local homotopies while retaining the same notion of cover, we introduce the corresponding undirected invariant. Definition 6.16. Let P : E → B be a functor. The strong normalized Švarc genus of P , denoted by Sg(P ), is the least integer n ≥ 0 for which B admits a cover {U0 , . . . , Un } such that, for every i, there exists a functor si : Ui → E for which P si is strongly homotopic to the inclusion ιi : Ui ,→ B. If no such finite cover exists, we set Sg(P ) = ∞. Thus, the local condition is the existence of a zigzag of natural transformations connecting P si and ιi , with no prescribed orientation. Remark 14. Definition 6.16 uses the morphism covers of Definition 4.1. It should therefore be distinguished from categorical versions of the Švarc genus defined using geometric covers [2, 4]. Proposition 6.17. For every functor P : E → B, Sg(P ) ≤ dsecat(P ).
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Proof. A local right homotopy section consists of a functor si : Ui → E and a natural transformation P si ⇒ ιi . This is, in particular, a zigzag of natural transformations between P si and ιi . Hence every cover admitting local right homotopy sections also admits local strong homotopy sections. □ Combining Proposition 6.17 with Proposition 6.4, we obtain Sg(P ) ≤ dsecat(P ) ≤ secat(P ). A Grothendieck bifibration is a functor that is both a Grothendieck fibration and a Grothendieck opfibration. In this case, the three invariants coincide. Proposition 6.18. Let P : E → B be a Grothendieck bifibration. Then Sg(P ) = dsecat(P ) = secat(P ). Proof. The strictification argument of [2, Proposition 2.18] shows that, for a bifibration, every local strong homotopy section can be replaced by a strict local section. The argument successively lifts the arrows in the zigzag, using the fibrational or opfibrational structure according to their orientation, until the homotopy section is replaced by a strict one. Although that result is stated using geometric covers, its proof applies independently to each member of the cover and therefore remains valid for the morphism covers used here. Hence Sg(P ) = secat(P ). Moreover, every Grothendieck bifibration is, in particular, a Grothendieck opfibration. Proposition 6.4 therefore gives dsecat(P ) = secat(P ), and the result follows.
□
6.7. Examples. Example 6.19. Let X : M → Set be a monoid action, as in Example 3.6. A cover of the one-object category M is precisely a family of submonoids whose union is M . Since πX is a discrete opfibration, its local right homotopy sections can be strictified. Consequently, dsecat(πX ) ≤ r − 1 if and only if there exist submonoids M1 , . . . , Mr ⊆ M and, for each i, an element xi ∈ X(∗) such that M = M1 ∪ · · · ∪ Mr
and
m · xi = xi
for every m ∈ Mi .
The elements xi may be different for different submonoids. Thus, dsecat(πX ) + 1 is the minimum number of submonoids covering M
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such that the action of each Mi has an Mi -fixed point. In particular, dsecat(πX ) = 0 if and only if the action of M has a global fixed point. The following example shows the difference between the directed sectional category and the strong Švarc genus. Example 6.20. Let P be the category of Example 5.17, generated by f1
x
g1
y
z g2
f2
subject to the relations (i, j, k, l ∈ {1, 2}).
gj fi = gl fk
Recall that dcat(P) = 1. Let D be the free category generated by x1
f11
y1 g1
x2
f12
f21
f22
y2
g2
z.
Define F : D → P by F (xi ) = x,
F (yi ) = y,
F (z) = z,
and F (fij ) = fj ,
F (gi ) = gi .
Since F is surjective on objects, Proposition 6.14 gives dsecat(F ) ≤ dcat(P) = 1. We claim that F has no global right homotopy section. Suppose otherwise that there exist a functor s : P → D and a natural transformation α : F s ⇒ 1P . Since x is a source, the component αx : F (s(x)) → x forces F (s(x)) = x. Hence s(x) = xi for some i ∈ {1, 2}, and αx = 1x . Naturality with respect to fk : x → y, for k = 1, 2, gives αy ◦ F (s(fk )) = fk . If F (s(y)) = x, then the existence of both morphisms s(fk ) : s(x) → s(y) forces s(y) = s(x) and s(fk ) = 1s(x) . The preceding equations would then give αy = f1 = f2 , a contradiction.
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If F (s(y)) = y, then s(y) = yj for some j ∈ {1, 2}, and αy = 1y . However, the unique morphism xi → yj is fij , whose image under F is fj . Thus both naturality conditions would require fj = f1
and
fj = f2 ,
again a contradiction. Finally, F (s(y)) = z is impossible because there is no morphism z → y in P. Therefore F has no global right homotopy section, so dsecat(F ) ̸= 0. Combining both inequalities yields dsecat(F ) = 1. On the other hand, let z : P → D be the constant functor at z. Then F z is the constant functor at the terminal object z of P. Hence there is a natural transformation 1P ⇒ F z. Thus F z is strongly homotopic to 1P , and Sg(F ) = 0. Consequently, Sg(F ) = 0 < dsecat(F ) = 1. 7. Databases and directed sectional category A database X : C → Set determines a discrete opfibration Z C X −→ C, πX : whose sections correspond to globally coherent choices of records. Since every discrete opfibration is a Grothendieck opfibration by Proposition 3.2, Proposition 6.4 gives dsecat(πX ) = secat(πX ). Thus every local right homotopy section of πX can be strictified. The directed formulation nevertheless remains useful, since it places the database problem within the homotopy-invariant framework developed in the preceding section, while the strict formulation gives its concrete interpretation in terms of coherent local selections. In this section, we study directed sectional category for the discrete opfibrations associated with functorial databases. We consider indecomposable and iterated databases, analyse its behaviour under data migration, and conclude by characterizing initial objects through the universal existence of global sections.
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7.1. Connected Components and Indecomposable Databases. Coproducts of databases are computed objectwise. A non-empty database X :` C → Set is called indecomposable if every decomposition X∼ X = 1 X2 has an empty summand. For this notion and its connections with categorical actions, Burnside rings, and biset functors, see [30]. The indecomposable summands of a database are determined by the connected components of its category of elements. Proposition 7.1. Let Z C X=
a
Eλ
λ∈Λ
be the decomposition of the category of elements of X into connected components. Then there are indecomposable subfunctors Xλ ⊆ X such that Z C a ∼ Xλ ∼ X= Xλ , = Eλ . λ∈Λ
Proof. For c ∈ C, define Xλ (c) = {x ∈ X(c) | (c, x) ∈ Eλ }. If f : c → d and x ∈ Xλ (c), the morphism (c, x) −→ d, X(f )(x)
shows that X(f )(x) ∈ Xλ (d). Hence Xλ is a subfunctor. Every ` object (c, x) belongs to a unique connected ` component, so ∼ X(c) = λ Xλ (c), naturally in c. Therefore X = λ Xλ , and by RC construction Xλ ∼ = Eλ . Since Eλ is connected, Xλ is indecomposable. □ Corollary 7.2. R C A non-empty database X : C → Set is indecomposable if and only if X is connected. For connected schemas, the existence of a global section can therefore be checked on the indecomposable summands. ` Proposition 7.3. Let C be connected, and let X ∼ = λ∈Λ Xλ be its decomposition into indecomposable summands. Then πX admits a global section if and only if πXλ admits a global section for some λ. RC Proof. The image of a section s : C → X is contained in a sinRC gle connected component, and hence s factors through some Xλ . The converse follows by composing a section of πXλ with the inclusion RC RC Xλ ,→ X. □
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Theorem 7.4. Let X∼ =
a
Xλ
λ∈Λ
be the decomposition of a database into indecomposable summands. Then, for every λ ∈ Λ, dsecat(πX ) ≤ dsecat(πXλ ). Consequently, if Λ is finite and non-empty, then dsecat(πX ) ≤ min dsecat(πXλ ). λ∈Λ
Proof. The inclusion Xλ ,→ X induces a functor Z C Z C Xλ −→ X jλ : over C. Since πX jλ = πXλ , Proposition 6.3 gives dsecat(πX ) ≤ dsecat(πXλ ). □ Remark 15. A local section of πX over a connected subcategory factors through a unique indecomposable summand. Different members of a cover may, however, use different summands. Therefore the preceding inequality need not be an equality, and dsecat(πX ) cannot in general be recovered from the directed sectional category of a single summand. 7.2. Iterated Databases. We now apply Proposition 6.3 to iterated functorial databases. A database on the category of elements of another database can be flattened into a single database on the original schema, and the corresponding category-of-elements projections agree under this identification. RC Let X : C → Set, and let Y : X → Set be a database on its category of elements. The standard equivalence Z C X, Set ≃ [C, Set]/X identifies Y with a database over C equipped with a natural transformation to X. Define the flattened database ΣX Y : C → Set by a (ΣX Y )(c) = Y (c, x). x∈X(c)
For f : c → d, set (ΣX Y )(f )(x, y) = X(f )(x), Y (f x )(y) ,
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where f x : (c, x) → (d, X(f )(x)) is the unique morphism over f in
RC
X.
Proposition 7.5. There is a canonical isomorphism Z RC X Z C ∼ Y = (ΣX Y ) under which πX ◦ πY corresponds to πΣX Y . Proof. Define Θ (c, x), y = c, (x, y) . The morphism conditions in the two categories agree by the definition of ΣX Y , so Θ is an isomorphism. Both projections send the corresponding object to c and the corresponding morphism to its underlying morphism in C. □ RC Corollary 7.6. For databases X : C → Set and Y : X → Set, dsecat(πX ) ≤ dsecat(πΣX Y ). Equivalently, secat(πX ) ≤ secat(πΣX Y ). Proof. Proposition 6.3 gives dsecat(πX ) ≤ dsecat(πX πY ). By Proposition 7.5, πX πY is identified with πΣX Y . The equivalent statement for secat follows because all the projections involved are discrete opfibrations. □ 7.3. Behaviour under Data Migration. Let F : C → D be a functor between database schemas. It induces three data migration functors ΣF ⊣ ∆F ⊣ ΠF [25]. Their types are ∆F : [D, Set] −→ [C, Set],
ΣF , ΠF : [C, Set] −→ [D, Set].
The functor ∆F is given by precomposition: ∆F Y = Y ◦ F for every database Y : D → Set. Thus, ∆F reindexes a database on D along F , retaining only the entity types and functional relationships visible through the schema C. The left and right migration functors are the left and right Kan extensions along F : ΣF = LanF ,
ΠF = RanF .
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Pointwise, for a database X : C → Set and an object d ∈ D, they are given by (ΣF X)(d) ∼ = colim X ◦ pd (F ↓d)
and (ΠF X)(d) ∼ = lim X ◦ qd , (d↓F )
where pd : (F ↓ d) −→ C,
qd : (d ↓ F ) −→ C
are the canonical projection functors. Hence ΣF transports data forward by collecting them through colimits, whereas ΠF transports data forward by imposing the compatibility conditions encoded by the corresponding limits. We now examine how these migrations interact with directed sectional category. We first study restriction along F , using the behaviour of directed sectional category under pullback. Proposition 7.7. Consider a pullback square F ∗E
Fe
E
F ∗P
P
C
F
D.
If P is a Grothendieck opfibration, then dsecat(F ∗ P ) ≤ dsecat(P ). Proof. Since P and F ∗ P are Grothendieck opfibrations, their directed sectional categories agree with their strict sectional categories. Let {U0 , . . . , Un } be a cover of D admitting local sections si : Ui → E of P . The inverse images F −1 (Ui ) form a cover of C, and every si pulls back to a section F ∗ si : F −1 (Ui ) −→ F ∗ E of F ∗ P . Hence dsecat(F ∗ P ) = secat(F ∗ P ) ≤ secat(P ) = dsecat(P ). □ As a corollary, we obtain the following. Corollary 7.8. Let F : C → D be a functor and let X : D → Set be a database. Then dsecat(π∆F X ) ≤ dsecat(πX ).
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Proof. There is a canonical pullback square RC RD e ∆F X F X π∆F X
πX
C
F
D,
where Fe(c, x) = (F (c), x). Thus π∆F X ∼ = F ∗ πX , and the result follows from Proposition 7.7. □ Thus, restriction along a schema functor cannot increase the sectional obstruction. This is consistent with the fact that ∆F may forget some of the compatibility conditions present in the original schema. The right migration functor has a particularly simple effect on global sections. Proposition 7.9. Let F : C → D be a functor and let X : C → Set be a database. There is a natural bijection Sect(πX ) ∼ = Sect(πΠ X ). F
Consequently, dsecat(πX ) = 0
⇐⇒
dsecat(πΠF X ) = 0.
Proof. Global sections of πX are naturally identified with natural transformations 1C ⇒ X. Using the adjunction ∆F ⊣ ΠF , we obtain Nat(1D , ΠF X) ∼ = Nat(∆F 1D , X). Since precomposition preserves the constant singleton database, ∆F 1D = 1C . Therefore, Nat(1D , ΠF X) ∼ = Nat(1C , X), which gives the required bijection. The final assertion follows because both projections are discrete opfibrations. □ The preceding proposition concerns only the existence of a global section. It does not, in general, imply that dsecat(πX ) and dsecat(πΠF X ) coincide when these invariants are strictly positive. 7.4. Detecting initial objects through databases. We conclude by characterizing directed category zero in terms of global sections of databases. A database X : C → Set is called objectwise non-empty if X(c) ̸= ∅ for every c ∈ C. Its category-of-elements projection is then surjective on objects. We first construct a canonical database that detects the existence of an initial object.
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Lemma 7.10. Let C be a finite connected acyclic category. Define XSrc : C → Set by a XSrc (c) = HomC (s, c), XSrc (u)(s, f ) = (s, u ◦ f ). s∈Src(C)
Then XSrc is objectwise non-empty, and πXSrc admits a global section if and only if C has an initial object. Proof. Every object c receives a morphism from some source: repeatedly follow incoming non-identity morphisms until the process terminates. Hence XSrc (c) ̸= ∅. Suppose that πXSrc admits a section. It corresponds to elements xc = (sc , fc ) ∈ XSrc (c) satisfying (sd , fd ) = (sc , u ◦ fc ) for every u : c → d. Thus sc = sd along every morphism. Since C is connected, there is a source i such that sc = i for every c. We have xi = (i, 1i ). If f : i → c, compatibility gives xc = (i, f ). Therefore any two morphisms i → c coincide, while xc = (i, fc ) provides such a morphism. Hence i is initial. Conversely, if i is initial and fc : i → c is the unique morphism, then xc = (i, fc ) defines a global section. □ Theorem 7.11. Let C be a finite connected acyclic category. The following are equivalent: (1) dcat(C) = 0; (2) every objectwise non-empty database X : C → Set admits a global section. Proof. If dcat(C) = 0, then Proposition 6.14 gives dsecat(πX ) = 0 for every objectwise non-empty X. Since πX is a discrete opfibration, Corollary 6.5 gives a global section. Conversely, apply the hypothesis to XSrc . Lemma 7.10 gives an initial object in C, and Proposition 5.2 yields dcat(C) = 0. □ Thus, for finite connected acyclic schemas, the vanishing of directed Lusternik–Schnirelmann category is equivalent to a universal solvability property for databases: every objectwise non-empty instance admits a globally coherent selection. References [1] Ernst Althaus, Benjamin Merlin Bumpus, James Fairbanks, Emilio Minichiello, and Daniel Rosiak. A parameterized algorithm for testing whether the limit of a diagram is empty, 2026. arXiv:2605.24240 [math.CT].
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[2] Isaac Carcacı́a-Campos, Enrique Macı́as-Virgós, and David Mosquera-Lois. Baues-Wirsching cohomology and švarc genus in small categories. Filomat, 40(8):3113–3132, 2026. [3] Isaac Carcacı́a-Campos, Enrique Macı́as-Virgós, and David Mosquera-Lois. Varadarajan’s theorem on categorical distance in small categories. Math. Slovaca, 76(3):835–845, 2026. [4] Isaac Carcacı́a-Campos, Enrique Macı́as-Virgós, and David Mosquera-Lois. Weak and strong fibrations of functors, 2026. arXiv:2605.18650 [math.CT]. [5] Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanré. LusternikSchnirelmann category, volume 103 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2003. [6] Sutirtha Datta, Navnath Daundkar, and Abhishek Sarkar. On the topological complexity of directed parametrized motion planning, 2025. arXiv:2504.06049 [math.AT]. [7] Matias L. del Hoyo. On the subdivision of small categories. Topology Appl., 155(11):1189–1200, 2008. [8] Jérémy Dubut, Éric Goubault, and Jean Goubault-Larrecq. The directed homotopy hypothesis. In Computer science logic 2016, volume 62 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 9, 16. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2016. [9] Lisbeth Fajstrup, Eric Goubault, Emmanuel Haucourt, Samuel Mimram, and Martin Raussen. Directed algebraic topology and concurrency. Springer, [Cham], 2016. With a foreword by Maurice Herlihy. [10] Brendan Fong and David I. Spivak. An invitation to applied category theory. Cambridge University Press, Cambridge, 2019. Seven sketches in compositionality. [11] Marco Grandis. The shape of a category up to directed homotopy. Theory Appl. Categ., 15:No. 4, 95–146, 2005/06. [12] Marco Grandis. Directed algebraic topology, volume 13 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2009. Models of nonreversible worlds. [13] Alexander Grothendieck. Revêtements étales et groupe fondamental. Fasc. I: Exposés 1 à 5. Institut des Hautes Études Scientifiques, Paris, 1963. Troisième édition, corrigée, Séminaire de Géométrie Algébrique, 1960/61. [14] Maximilian Hadek, Tomáš Jakl, and Jakub Opršal. A categorical perspective on constraint satisfaction: The wonderland of adjunctions, 2026. arXiv:2503.10353 [cs.LO]. [15] Dmitry Kozlov. Combinatorial algebraic topology, volume 21 of Algorithms and Computation in Mathematics. Springer, Berlin, 2008. [16] Ming Jung Lee. Homotopy for functors. Proc. Amer. Math. Soc., 36:571–577; erratum, ibid. 42 (1973), 648–650, 1972. [17] Fosco Loregian and Emily Riehl. Categorical notions of fibration. Expo. Math., 38(4):496–514, 2020. [18] Enrique Macı́as-Virgós and David Mosquera-Lois. Homotopic distance between functors. J. Homotopy Relat. Struct., 15(3-4):537–555, 2020. [19] Elias Gabriel Minian. Cat as a Λ-cofibration category. J. Pure Appl. Algebra, 167(2-3):301–314, 2002.
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[20] Ieke Moerdijk. Classifying spaces and classifying topoi, volume 1616 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1995. [21] Jacob Neumann and Thorsten Altenkirch. Synthetic 1-categories in directed type theory. In 30th International Conference on Types for Proofs and Programs, volume 336 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 7, 23. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2025. [22] nLab authors. nerve. https://ncatlab.org/nlab/show/nerve, July 2026. Revision 86. [23] Daniel Quillen. Higher algebraic K-theory. I. In Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), volume Vol. 341 of Lecture Notes in Math., pages 85–147. Springer, Berlin-New York, 1973. [24] Giuseppe Rosolini. About Grothendieck fibrations. In The mathematical and philosophical legacy of Alexander Grothendieck, Chapman Math. Notes, pages 229–264. Birkhäuser/Springer, Cham, [2025] ©2025. [25] David I. Spivak. Functorial data migration. Inform. and Comput., 217:31–51, 2012. [26] David I. Spivak. Database queries and constraints via lifting problems. Math. Structures Comput. Sci., 24(6):e240602, 55, 2014. [27] Ross Street and R. F. C. Walters. The comprehensive factorization of a functor. Bull. Amer. Math. Soc., 79:936–941, 1973. [28] Kohei Tanaka. Lusternik-Schnirelmann category for categories and classifying spaces. Topology Appl., 239:65–80, 2018. [29] Robert Wayne Thomason. Cat as a closed model category. Cahiers Topologie Géom. Différentielle, 21(3):305–324, 1980. [30] Peter Webb. Biset functors for categories, 2023. arXiv:2304.06863 [math.RT]. Isaac Carcacı́a-Campos, Departamento de Matemáticas, Universidade de Santiago de Compostela, 15782-SPAIN Email address: [email protected]