Using the Chu construction for generalizing formal concept analysis

Abstract

The goal of this paper is to show a connection between FCA generalisations and the Chu construction on the category ChuCors, the category of formal contexts and Chu correspondences. All needed categorical properties like categorical product, tensor product and its bifunctor properties are presented and proved. Finally, the second order generalisation of FCA is represented by a category built up in terms of the Chu construction.

Keywords

• formal concept analysis
• category theory
• Chu construction

Introduction

The importance of category theory as a foundational tool was discovered soon after its very introduction by Eilenberg and MacLane about seventy years ago. On the other hand, Formal Concept Analysis (FCA) has largely shown both its practical applications and its capability to be generalized to more abstract frameworks, and this is why it has become a very active research topic in the recent years; for instance, a framework for FCA has been recently introduced in [ 19 ] in which the sets of objects and attributes are no longer unstructured but have a hypergraph structure by means of certain ideas from mathematical morphology. On the other hand, for an application of the FCA formalism to other areas, in [ 11 ] the authors introduce a representation of algebraic domains in terms of FCA.

The Chu construction [ 8 ] is a theoretical method that, from a symmetric monoidal closed (autonomous) category and a dualizing object, generates a *autonomous category. This construction, or the closely related notion of Chu space, has been applied to represent quantum physical systems and their symmetries [ 1, 2 ].

This paper continues with the study of the categorical foundations of formal concept analysis. Some authors have noticed the property of being a cartesian closed category of certain concept structures that can be approximated [ 10, 20 ]; others have provided a categorical construction of certain extensions of FCA [ 12 ]; morphisms have received a categorical treatment in [ 17 ] as a means for the modelling of communication.

There already exist some approaches [ 9 ] which consider the Chu construction in terms of FCA. In the current paper, we continue the previous study by the authors on the categorical foundation of FCA [ 13,15,16 ]. Specifically, the goal of this paper is to highlight the importance of the Chu construction in the research area of categorical description of the theory of FCA and its generalisations. The Chu construction plays here the role of some recipe for constructing a suitable category that covers the second order generalisation of FCA.

The structure of this paper is the following: in Section 2 we recall the preliminary notions required both from category theory and formal concept analysis. Then, the various categorical properties of the input category which are required (like the existence of categorical and tensor product) are developed in detail in Sections 3 and 4. An application of the Chu construction is presented in Section 5 where it is also showed how to construct formal contexts of second order from the category of classical formal contexts and Chu correspondences (ChuCors). 2

Preliminaries

In order to make the manuscript self-contained, the fundamental notions and its required properties are recalled in this section.

Definition 1. A formal context is any triple C = hB, A, Ri where B and A are finite sets and R ⊆ B × A is a binary relation. It is customary to say that B is a set of objects, A is a set of attributes and R represents a relation between objects and attributes.

On a given formal context (B, A, R), the derivation (or concept-forming) operators are a pair of mappings ↑ : 2B → 2A and ↓ : 2A → 2B such that if X ⊆ B, then ↑X is the set of all attributes which are related to every object in X and, similarly, if Y ⊆ A, then ↓Y is the set of all objects which are related to every attribute in Y .

In order to simplify the description of subsequent computations, it is convenient to describe the concept forming operators in terms of characteristic functions, namely, considering the subsets as functions on the set of Boolean values. Specifically, given X ⊆ B and Y ⊆ A, we can consider mappings ↑X : A → {0, 1} and ↓Y : B → {0, 1} 1. ↑X(a) = ^ (b ∈ X) ⇒ ((b, a) ∈ R) for any a ∈ A

b∈B 2. ↓Y (b) = ^ (a ∈ Y ) ⇒ ((b, a) ∈ R) for any b ∈ B

a∈A where the infimum is considered in the set of Boolean values and ⇒ is the truthfunction of the implication of classical logic. Definition 2. A formal concept is a pair of sets hX, Y i ∈ 2B × 2A which is a fixpoint of the pair of concept-forming operators, namely, ↑X = Y and ↓Y = X. The object part X is called the extent and the attribute part Y is called the intent.

There are two main constructions relating two formal contexts: the bonds and the Chu correspondences. Their formal definitions are recalled below: Definition 3. Consider C1 = hB1, A1, R1i and C2 = hB2, A2, R2i two formal contexts. A bond between C1 and C2 is any relation β ∈ 2B1×A2 such that its columns are extents of C1 and its rows are intents of C2. All bonds between such contexts will be denoted by Bonds(C1, C2).

The Chu correspondence between contexts can be seen as an alternative inter-contextual structure which, instead, links intents of C1 and extents of C2. Namely, Definition 4. Consider C1 = hB1, A1, R1i and C2 = hB2, A2, R2i two formal contexts. A Chu correspondence between C1 and C2 is any pair of multimappings ϕ = hϕL, ϕRi such that – ϕL : B1 → Ext(C2) – ϕR : A2 → Int(C1) – ↑2(ϕL(b1))(a2) = ↓1(ϕR(a2))(b1) for any (b1, a2) ∈ B1 × A2 All Chu correspondences between such contexts will be denoted by Chu(C1, C2).

The notions of bond and Chu correspondence are interchangeable; specifically, we will use the bond βϕ associated to a Chu correspondence ϕ from C1 to C2 defined for b1 ∈ B1, a2 ∈ A2 as follows:

βϕ(b1, a2) = ↑2 (ϕL(b1))(a2) = ↓1 (ϕR(a2))(b1)

The set of all bonds (resp. Chu correspondences) between any two formal contexts endowed with set inclusion as ordering have a complete lattice structure. Moreover, both complete lattices are dually isomorphic.

In order to formally define the composition of two Chu correspondences, we need to introduce the extension principle below: Definition 5. Given a mapping ϕ : X → 2Y we define its extended mapping ϕ+ : 2X → 2Y defined by ϕ+(M ) = Sx∈M ϕ(x), for all M ∈ 2X .

The set of formal contexts together with Chu correspondences as morphisms forms a category denoted by ChuCors. Specifically: – objects formal contexts – arrows Chu correspondences – identity arrow ι : C → C of context C = hB, A, Ri • ιL(o) = ↓↑ ({b}), for all b ∈ B • ιR(a) = ↑↓ ({a}), for all a ∈ A – composition ϕ2 ◦ ϕ1 : C1 → C3 of arrows ϕ1 : C1 → C2, ϕ2 : C2 → C3 (where Ci = hBi, Ai, Rii, i ∈ {1, 2, 3}) • (ϕ2 ◦ ϕ1)L : B1 → 2B3 and (ϕ2 ◦ ϕ1)R : A3 → 2A1 • (ϕ2 ◦ ϕ1)L(b1) = ↓3↑3 (ϕ2L+(ϕ1L(b1))) • (ϕ2 ◦ ϕ1)R(a3) = ↑1↓1 (ϕ1R+(ϕ2R(a3)))

The category ChuCors is *-autonomous and equivalent to the category of complete lattices and isotone Galois connection, more results on this category and its L-fuzzy extensions can be found in [ 13, 15, 16, 18 ]. 3

Categorical product on ChuCors

In this section, the category ChuCors is proved to contain all finite categorical products, that is, it is a Cartesian category. To begin with, it is convenient to recall the notion of categorical product.

Definition 6. Let C1 and C2 be two objects in a category. By a product of C1 and C2 we mean an object P with arrows πi : P → Ci for i ∈ {1, 2} satisfying the following condition: For any object D and arrows δi : D → Ci for i ∈ {1, 2}, there exists a unique arrow γ : D → P such that γ ◦ πi = δi for all i ∈ {1, 2}.

The construction will use the notion of disjoint union of two sets S1 ] S2 which can be formally described as ({1} × S1) ∪ ({2} × S2) and, therefore, their elements will be denoted as ordered pairs (i, s) where i ∈ {1, 2} and s ∈ Si. Now, we can proceed with the construction: Definition 7. Consider C1 = hB1, A1, R1i and C2 = hB2, A2, R2i two formal contexts. The product of such contexts is a new formal context

C1 × C2 = hB1 ] B2, A1 ] A2, R1×2i where the relation R1×2 is given by

((i, b), (j, a)) ∈ R1×2 if and only if (i = j) ⇒ (b, a) ∈ Ri for any (b, a) ∈ Bi × Aj and (i, j) ∈ {1, 2} × {1, 2}.

Lemma 1. The above defined contextual product fulfills the property of the categorical product on the category ChuCors.

Proof. We define the projection arrows hπiL, πiRi ∈ Chu(C1×C2, Ci) for i ∈ {1, 2} as follows –– ππiiLR :: BA1i →]BIn2t→(C1E×xtC(C2)i)⊆⊆22AB1∪iA2 – such that for any (k, x) ∈ B1 ] B2 and ai ∈ Ai the following equality holds ↑i (πiL(k, x))(ai) = ↓1×2 (πiR(ai))(k, x) The definition of the projections is given below πiL(k, x)(bi) = πiR(ai)(k, y) = ( ↓i↑i (χx)(bi) for k = i

↓i↑i 0 (bi) for k 6= i ( ↑i↓i (χai )(y) for k = i

↑k↓k 0 (y) for k 6= i

The proof that the definitions above actually provide a Chu correspondence is just a long, although straightforward, computation and it is omitted.

Now, one has to show that to any formal context D = hE, F, Gi, where G ⊆ E × F and any pair of arrows (δ1, δ2) with δi : D → Ci for all i ∈ {1, 2}, there exists a unique morphism γ : D → C1 × C2 such that the following diagram commutes: for any (k, y) ∈ A1 ] A2 and ai ∈ Ai.

for any (k, x) ∈ B1 ] B2 and bi ∈ Bi We give just the definition of γ as a pair of mappings γL : E → 2B1]B2 and γR : A1 ] A2 → 2F – γL(e)(k, x) = δkL(e)(x) for any e ∈ E and (k, x) ∈ B1 ] B2. – γR(k, y)(f ) = δkR(y)(f ) for any f ∈ F and (k, y) ∈ A1 ] A2.

Checking the condition of categorical product is again straightforward but long and tedious and, hence, it is omitted. tu

We have just proved that binary products exist, but a cartesian category requires the existence of all finite products. If we recall the well-known categorical theorem which states that if a category has a terminal object and binary product, then it has all finite products, we have just to prove the existence of a terminal object (namely, the nullary product) in order to prove ChuCors to be cartesian.

Any formal context of the form hB, A, B × Ai where the incidence relation is the full cartesian product of the sets of objects and attributes is (isomorphic to) the terminal object of ChuCors. Such formal context has just one formal concept hB, Ai; hence, from any other formal context there is just one Chu correspondence to hB, A, B × Ai. 4

Tensor product and its bifunctor property

Apart from the categorical product, another product-like construction can be given in the category ChuCors, for which the notion of transposed context C∗ is needed.

Given a formal context C = hB, A, Ri, its transposed context is C∗ = hA, B, Rti, where Rt(a, b) holds iff R(b, a) holds. Now, if ϕ ∈ Chu(C1, C2), one can consider ϕ∗ ∈ Chu(C2∗, C1∗) defined by ϕ∗L = ϕR and ϕ∗R = ϕL.

Definition 8. The tensor product of formal contexts Ci = hBi, Ai, Rii for i ∈ {1, 2} is defined as the formal context C1 C2 = hB1 ×B2, Chu(C1, C2∗), R i where

R ((b1, b2), ϕ) = ↓2 (ϕL(b1))(b2).

Mori studied in [ 18 ] the properties of the tensor product above, and proved that ChuCors with is a symmetric and monoidal category. Those results were later extended to the L-fuzzy case in [ 13 ]. In both papers, the structure of the formal concepts of a product context was established as an ordered pair formed by a bond and a set of Chu correspondences.

Lemma 2. Let Ci = hBi, Ai, Rii for i ∈ {1, 2} be two formal contexts, and let hβ, Xi ∈ Bonds(C1, C2∗) × 2Chu(C1,C2∗) be an arbitrary formal concept of C1 C2. Then β = Vψ∈X βψ and X = {ψ ∈ Chu(C1, C2∗) | β ≤ βψ}.

Proof. Let X be an arbitrary subset of Chu(C1, C2∗). Then, for all (b1, b2) ∈ B1 × B2, we have

^ ↓C1 C2 (X)(b1, b2) =

(ψ ∈ X) ⇒ ↓2 (ψL(b1))(b2) ψ∈Chu(C1,C2∗) = ^ ↓2 (ψL(b1))(b2) = ψ∈X ^ βψ(b1, b2) ψ∈X Let β be an arbitrary subset of B1 × B2. Then, for all ψ ∈ Chu(C1, C2∗) ↑C1 C2 (β)(ψ) = =

^ (b1,b2)∈B1×B2

^ (b1,b2)∈B1×B2 β(b1, b2) ⇒ ↓2 (ψL(b1))(b2) β(b1, b2) ⇒ βψ(b1, b2) Hence ↑C1 C2 (β) = {ψ ∈ Chu(C1, C2∗) | β ≤ βψ} tu

We now introduce the notion of product of one context with a Chu correspondence.

Definition 9. Let Ci = hBi, Ai, Rii for i ∈ {0, 1, 2} be formal contexts, and consider ϕ ∈ Chu(C1, C2). Then, the pair of mappings (C0 ϕ)L : B0 × B1 → 2B0×B2 (C0 ϕ)R : Chu(C0, C2) → 2Chu(C0,C1) is defined as follows: – (C0 ϕ)L(b, b1)(o, b2) = ↓C0 C2 ↑C0 C2 (γϕb,b1 )(o, b2) where

γϕb,b1 (o, b2) = (b = o) ∧ ϕL(b1)(b2) for any b, o ∈ B0, bi ∈ Bi with i ∈ {1, 2} – (C0 ϕ)R(ψ2)(ψ1) = ψ1 ≤ (ψ2 ◦ ϕ∗) for any ψi ∈ Chu(C0, Ci)

As one could expect, the result is a Chu correspondence between the products of the contexts. Specifically: Lemma 3. Let Ci = hBi, Ai, Rii be formal contexts for i ∈ {0, 1, 2}, and consider ϕ ∈ Chu(C1, C2). Then C0 ϕ ∈ Chu(C0 C1, C0 C2).

Proof. (C0 ϕ)L(b, b1) ∈ Ext(C0 C2) for any (b, b1) ∈ B0 × B1 follows directly from its definition. (C0 ϕ)R(ψ) ∈ Int(C0 C1) for any ψ ∈ Chu(C0, C1) follows from Lemma 2.

Consider an arbitrary b ∈ B0, b1 ∈ B1 and ψ2 ∈ Chu(C0, C2∗) ↑C0 C2 (C0

ϕ)L(b, b1) (ψ2) = ↑C0 C2 ↓C0 C2 ↑C0 C2 (γϕb,b1 )(ψ2) = ↑C0 C2 (γϕb,b1 )(ψ2) ^

γϕb,b1 (o, b2) ⇒ ↓ (ψ2R(b2))(o) = = (o,b2)∈B0×B2

^ (o,b2)∈B0×B2 = ^ ^

o∈B0 b2∈B2 = ^ (o = b) ⇒

o∈B0 = ^

b2∈B2 = ^

b2∈B2 = ^ a∈A ϕL(b1)(b2) ⇒ ↓ (ψ2R(b2))(b) ϕL(b1)(b2) ⇒ ^ (ψ2R(b2)(a) ⇒ R(b, a)) a∈A = ^

_ (ϕL(b1)(b2) ∧ ψ2R(b2)(a)) ⇒ R(b, a) a∈A b2∈B2 ψ2R+(ϕL(b1))(a) ⇒ R(b, a)

(o = b) ∧ ϕL(b1)(b2) ⇒ ↓ (ψ2R(b2))(o) (o = b) ⇒

ϕL(b1)(b2) ⇒ ↓ (ψ2R(b2))(o) ^ (ϕL(b1)(b2) ⇒ ↓ (ψ2R(b2))(o)) b2∈B2 = ↓ (ψ2R+(ϕL(b1))(b) = ↓↑↓ (ψ2R+(ϕL(b1))(b) = ↓ ((ϕ ◦ ψ2)R(b1))(b) Note the use above of the extended mapping as given in Definition 5 in relation to the composition of Chu correspondences.

On the other hand, we have ↓C0 C1 ((C0 ϕ)R(ψ2))(b, b1)

^ = = ψ1∈Chu(C0,C1)

^ ψ1∈Chu(C0,C1) ((C0

ϕ)R(ψ2)(ψ1) ⇒ ↓ (ψ1R(b1))(b)) ((ψ1 ≥ ϕ ◦ ψ2) ⇒ ↓ (ψ1R(b1))(b))

^

↓ (ψ1R(b1))(b) = ↓ ((ϕ ◦ ψ2)R(b1))(b) tu

Given a fixed formal context C, the tensor product C (−) forms a mapping between objects of ChuCors assigning to any formal context D the formal context C D. Moreover to any arrow ϕ ∈ Chu(C1, C2) it assigns an arrow C ϕ ∈ Chu(C C1, C C2). We will show that this mapping preserves the unit arrows and the composition of Chu correspondences. Hence the mapping forms an endofunctor on ChuCors, that is, a covariant functor from the category ChuCors to itself.

To begin with, let us recall the definition of functor between two categories: Definition 10 (See [ 6 ]). A covariant functor F : C → D between categories C and D is a mapping of objects to objects and arrows to arrows, in such a way that: – For any morphism f : A → B, one has F (f ) : F (A) → F (B) – F (g ◦ f ) = F (g) ◦ F (f ) – F (1A) = 1F (A).

Lemma 4. Let C = hB, A, Ri be a formal context. C on ChuCors. (−) is an endofunctor Proof. Consider the unit morphism ιC1 of a formal context C1 = hB1, A1, R1i, and let us show that (C ιC1 ) = ιC C1 . In other words, C (−) respects unit arrows in ChuCors.

↑C C1 (C ιC1 )(b, b1) (ψ)

^ =

(o,o1)∈B×B1 = ^

o1∈B1 = ^

o1∈B1 = ^

^ o1∈B1 a1∈A1 = ^ ^

o1∈B1 a1∈A1 = ^

a1∈A1 = ^ o1∈B1 (o = b) ∧ ιC1L(b1)(o1) ⇒ ↓1 (ψL(o))(o1) ↓1↑1 (χb1 )(o1) ⇒ ↓1 (ψL(b))(o1) ↓1↑1 (χb1 )(o1) ⇒

ψL(b)(a1) ⇒ R(o1, a1) ↓1↑1 (χb1 )(o1) ⇒

ψL(b)(a1) ⇒ R(o1, a1) ψL(b)(a1) ⇒

↓1↑1 (χb1 )(o1) ⇒ R(o1, a1) ψL(b)(a1) ⇒

↓1↑1 (χb1 )(o1) ⇒ R(o1, a1) ψL(b)(a1) ⇒ ↑1↓1↑1 (χb1 )(a1) = ^

ψL(b)(a1) ⇒ R1(b1, a1) = ↓1 (ψL(b))(b1) and, on the other hand, we have ↑C C1 (ιC C1 (b, b1))(ψ) = = ↑C C1 (χ(b,b1))(ψ)

^ (o,o1)∈B×B1 = ↓1 (ψL(b))(b1)

χ(b,b1)(o, o1) ⇒ ↓1 (ψL(o))(o1) As a result, we have obtained ↑C C1 ((C ιC1 )(b, b1))(ψ) =↑C C1 (ιC C1 (b, b1))(ψ) for any (b, b1) ∈ B × B1 and any ψ ∈ Chu(C, C1); hence, ιC C1 = (C ιC1 ).

We will show now that C (−) preserves the composition of arrows. Specifically, this means that for any two arrows ϕi ∈ Chu(Ci, Ci+1) for i ∈ {1, 2} it holds that C (ϕ1 ◦ ϕ2) = (C ϕ1) ◦ (C ϕ2).

↑C C3 C (ϕ1 ◦ ϕ2) L(b, b1) (ψ3) ^ =

(o,b3)∈B×B3 = ^ = ↓ (ϕ1 ◦ ϕ2 ◦ ψ3)L(b1) (b) On the other hand, and writing F for C expressions, we have ↑F C3 ((F ϕ1 ◦ F ϕ2)L(b, b1))(ψ3) = ↑F C3 ↓F C3 ↑F C3 (F ϕ2)L+ (F ϕ1)L(b, b1) (ψ3)

(o = b) ∧ (ϕ1 ◦ ϕ2)L(b1)(b3) ⇒ ↓ (ψ3R(b3))(o) (ϕ1 ◦ ϕ2)L(b1)(b3) ⇒ ↓ (ψ3R(b3))(b) b3∈B3 (by similar operations to those in the first part of the proof) − in order to simplify the resulting =

^ (o,b3)∈B×B3

_ (j,b2)∈B×B2 = ^ ^

b3∈B3 b2∈B2 = ^

_ b3∈B3 b2∈B2 = ^ (F ϕ1)L(b, b1)(j, b2) ∧ (F ϕ2)L(j, b2)(o, b3) ⇒ ↓ (ψ3R(b3))(o) ϕ1L(b1)(b2) ∧ ϕ2L(b2)(b3) ⇒ ↓ (ψ3R(b3))(b) ϕ1L(b1)(b2) ∧ ϕ2L(b2)(b3) ⇒ ↓ (ψ3R(b3))(b) ϕ2L+(ϕ1L(b1))(b3) ⇒ ↓ (ψ3R(b3))(b)

= ^ b3∈B3 Proposition 1. The tensor product forms a bifunctor − ChuCors to ChuCors. − from ChuCors × 5

The Chu construction on ChuCors and second order formal concept analysis

A second order formal context [ 14 ] focuses on the external formal contexts and it serves a bridge between the L-fuzzy [ 3, 7 ] and heterogeneous [ 4 ] frameworks. Definition 11. Consider two non-empty index sets I and J and an L-fuzzy formal context h i∈I Bi, Sj∈J Aj, ri, whereby

S – Bi1 ∩ Bi2 = ∅ for any i1, i2 ∈ I, – Aj1 ∩ Aj2 = ∅ for any j1, j2 ∈ J , – r : Si∈I Bi × Sj∈J Aj −→ L.

Moreover, consider two non-empty sets of L-fuzzy formal contexts (external formal contexts) notated by – {hBi, Ti, pii : i ∈ I}, whereby Ci = hBi, Ti, pii, – {hOj, Aj, qji : j ∈ J }, whereby Dj = hOj, Aj, qji.

A second order formal context is a tuple

D [ Bi, {Ci; i ∈ I}, [ Aj, {Dj; j ∈ J }, i∈I j∈J [ (i,j)∈I×J

E ri,j , whereby ri,j : Bi × Aj −→ L is defined as ri,j(o, a) = r(o, a) for any o ∈ Bi and a ∈ Aj.

The Chu construction [ 8 ] is a theoretical process that, from a symmetric monoidal closed (autonomous) category and a dualizing object, generates a *autonomous category. The basic theory of *-autonomous categories and their properties are given in [ 5, 6 ].

In the following, the construction will be applied on ChuCors and the dualizing object ⊥ = h{ }, { }, 6=i as inputs. In this section it is shown how second order FCA [ 14 ] is connected to the output of such construction.

The category generated by the Chu construction and ChuCors and ⊥ will be denoted by CHU(ChuCors, ⊥): – Its objects are triplets of the form hC, D, ρi where • C and D are objects of the input category ChuCors (i.e. formal contexts) • ρ is an arrow in Chu(C D, ⊥) – Its morphisms are pairs of the form hϕ, ψi : hC1, C2, ρ1i → hD1, D2, ρ2i where Ci and Di are formal contexts for i ∈ {1, 2} and • ϕ and ψ are elements from Chu(C1, D1) and Chu(D2, C2), respectively, such that the following diagram commutes ϕ

C1 D2 D1 ∨

D2 D2

C1 ψ > C1 ρ2

∨ > ⊥

C2 ρ1 or, equivalently, the following equality holds (C1 ψ) ◦ ρ1 = (ϕ

D2) ◦ ρ2

There are some interesting facts in the previous construction with respect to the second order FCA [ 14 ]: 1. To begin with, every object hC1, C2, ρi in CHU(ChuCorsL, ⊥), and recall that ρ ∈ Chu(C1 C2, ⊥), can be represented as a second order formal context (from Definition 11). Simply take into account that, from basic properties of the tensor product, we can obtain Chu(C1 C2, ⊥) ∼= Chu(C1, C2∗). Specifically, as ChuCors is a closed monoidal category, we have that for every three formal contexts C1, C2, C3 the following isomorphism holds ChuCors(C1

C2, C3) =∼ ChuCors(C1, C2 ( C3), whereby C2 ( C3 denotes the value at C3 of the right adjoint and recall that C2 ( ⊥ =∼ C2∗ because ChuCors is *-autonomous. The other necessary details about closed monoidal categories and the corresponding notations one can find in [ 6 ]. 2. Similarly, any second order formal context (from Definition 11) is representable by an object of CHU(ChuCorsL, ⊥). 6

Conclusions and future work

After introducing the basic definitions needed from category theory and formal concept analysis, in this paper we have studied two different product constructions in the category ChuCors, namely the categorical product and the tensor product. The existence of products allows to represent tables and, hence, binary relations; the tensor product is proved to fulfill the required properties of a bifunctor, which enables us to consider the Chu construction on the category ChuCors. As a first application, we have sketched the representation of second order formal concept analysis [ 14 ] in terms of the Chu construction on the category ChuCors.

The use of different subcategories of ChuCors as input to the Chu construction seems to be an interesting way of obtaining different existing generalizations of FCA. For future work, we are planning to provide representations based on the Chu construction for one-sided FCA, heterogeneous FCA, multi-adjoint FCA, etcetera.

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