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On the category of structure species

The purpose of the present paper is to make a mathematical study of the differences and relations among possible structures inherent in an object, as well as of the whole structure constituted by them (i.e., the structure of structures), against the background of the structuralism by Claude L\'{e}vi-Strauss and others. Our discussion focuses on Blanchard's categorical reformulation of the notion of structure species introduced originally by Bourbaki. The main result of the present paper asserts that a category can be reconstructed, up to a certain slight indeterminacy, from the category of structure species on it. This result is partially motivated by various reconstruction theorems that have been shown in the context of anabelian geometry.

1. Introduction: Lévi-Strauss' structuralism 1.1. Structuralism is a general theory of culture and methodology that focuses on relationships rather than individual objects, or alternatively, where objects are defined by the set of relationships of which they are part and not by the qualities possessed by them taken in isolation. According to this mode of knowledge, phenomena of human life are not intelligible except through their interrelations. These relations constitute a structure, and behind local variations in the surface phenomena there are constant laws of abstract structure (cf. [Cal], [Rot], [SBla]).
Structuralism in Europe developed in the early 20th century from insights in the field of linguistics of, mainly, Ferdinand de Saussure and the subsequent Prague, Moscow, and Copenhagen schools of linguistics. After World War II, an array of scholars in the humanities borrowed Saussure's concepts for use in their respective fields. By the early 1960s, structuralism as a movement was coming into its own and some believed that it offered a single unified approach to human life that would embrace all disciplines. Claude Lévi-Strauss, a French anthropologist, was the first such scholar, sparking a widespread interest in structuralism.
Lévi-Strauss defined "structure" as a whole consisting of elements and relations between elements, which retain their invariant properties through a series of transformations. To be precise, a model with structural value satisfies several requirements described as follows (cf. [Lev2], Part 5, Chap. XV): (i) The structure exhibits the characteristics of a system. It is made up of several elements, none of which can undergo a change without effecting changes in all the other elements. (ii) For any given model there should be a possibility of ordering a series of transformations resulting in a group of models of the same type. (iii) The above properties make it possible to predict how the model will react if one or more of its elements are submitted to certain modifications. (iv) The model should be constituted so as to make immediately intelligible all the observed facts. The structures listed, at least, as illustrations are kinships, political ideologies, mythologies, ritual, art, code of etiquette, and even cooking (cf. [Rot]). The discernment of these structures and their comparative analysis, which takes into account their distribution both historically and geographically, is indeed the subject matter of structural anthropology. In "The Elementary Structures of Kinship" (cf. [Lev3]), Claude Lévi-Strauss examined kinship systems from a structural point of view and demonstrated how apparently different social organizations were different permutations of a few basic kinship structures. The kinship system provides a way to order individuals according to certain rules; social organization is another way of ordering individuals and groups; social stratifications, whether economic or political, provide us with a third type; and all these orders can themselves be ordered by showing the kinds of relationship which exist among them, i.e., how they interact with one another on both the synchronic and the diachronic levels. One may construct models valid not only for one type of order (kinship, social organization, economic relations, etc.) but where numerous models for all types of order are themselves ordered inside a total model. In [Rot], Chap. II, N. Rotenstreich explained that We do not start out with scattered concepts: we start with structures and move to further structures. We start with order and move from another order or to an order of orders. There is a structure to the relation between these orders or structures. This "structure of structures" * is not just a static relation of coexistence, i.e. language beside kinship, etc., or even not one of subordination whereby a narrow structure such as rites is comprised in or is secondary to a wider structure such as society.
1.2. In the present paper, we make a mathematical study of the differences and relations among possible structures inherent in an object, as well as the whole structure constituted by them (i.e., the structure of structures), against the background of the structuralism by Lévi-Strauss and others. Our discussion focuses on some objects treated in structural theory, starting with Bourbaki. As is well known, Bourbaki, the collective pseudonym of a group of predominantly French mathematicians, undertook, in the mid-20th century, the task of making a unified development of central parts of modern mathematics in largely formalized language. † This resulted in a long series of books "Éléments de Mathématique" that became very influential. In a manifesto written by Bourbaki in 1950, some main principles of their structuralist view of mathematics were presented. In one of these books (cf. [Bou]), Bourbaki developed their theory of structures in a settheoretic manner; the building blocks are called structure species. Roughly speaking, a structure species is a set, or a collection of sets, endowed with relations and operations not only among their members but also among collections of elements of these sets, relations among them, etc. A basic example is the structure species of ordered sets, where, from a set S, we obtain (by a suitable echelon construction scheme) the power set P (S × S) of the product S × S and a binary relation s ∈ P (S × S) (called the typical characterization) equipped with a relation describing the axiom of an ordered set.
In [ABla], Blanchard introduced the concept of structure species on a category very close to the concept of structure species in the sense of Bourbaki and proved that it is equivalent to the concept of structure species in the sense of Sonner (cf. [Son]). The purpose of the present paper is to consider "structure of structures" formulated in terms of Blanchard's structure species and investigate how this concept can capture the essence of things. That is, we discuss the issue of how much information concerning a given category is contained in the knowledge about the structure of structure species on that category.
The following is the main result of the present paper, which asserts that a category can be reconstructed, up to a certain slight indeterminacy, from the category of structure species on it (the proof will be given in § 3): Theorem A. Let us fix a universe U . Let X and X ′ be connected U -small categories. Denote by Sp X and Sp X ′ the categories of structure species on X and X ′ respectively. Also, denote by X op the opposite category of X. Then, the following two conditions are equivalent to each other: 1.3. Finally, we remark that Theorem A can be regarded as a variant of various reconstruction theorems that have been shown in the context of anabelian geometry (cf. Remark 3.4.2). In 1983, A. Grothendieck wrote a letter to G. Faltings (cf. [Gro2]), outlining what is today known as the anabelian conjectures. (Many of the claims based on these conjectures have now been proved by mathematicians.) These conjectures concern the possibility of reconstructing certain arithmetic varieties (e.g., hyperbolic curve over a number or a p-adic local field) from theirétale fundamental groups. Our study is partially motivated by the anabelian philosophy of Grothendieck; this is because a structure species, or equivalently a constructive functor (cf. Definition 2.3.1, (i)), on a category X may be regarded, in some sense, as categorical realization of coverings over X (cf. [Son], § 3). Moreover, if X is a groupoid, then one can interpret Theorem A as a reconstruction assertion for X by means of the fundamental group associated with the Galois category of constructive functors over X (cf. Remark 2.5.3). Similar category-theoretic reconstructions can be found in [Moc1], [Moc2], and [Wak].
rules for four classes of people within Australian aboriginal society. This contribution appeared in an appendix of Lévi-Strauss's book (cf. [Lev3]).
Acknowledgements. The author was partially supported by Grant-in-Aid for Scientific Research (KAKENHI No. 18K13385, 21K13770).

Structures species and constructive functors
In this section, we recall structure species on a category defined by Blanchard and the equivalence of structure species and constructive functors. After that, we examine constructive functors on a groupoid.
2.1. Preliminaries on categories. Let C be a category. We denote by Ob(C) (resp., Ob(C); resp., Mor(C)) the set of objects (resp., the set of isomorphism classes of objects; resp., the set of morphisms) in C. For two objects a, b ∈ Ob(C), we denote by Mor C (a, b), or Mor(a, b), the set of morphisms a → b in C. Also, we write (hence Mor{a, b} = Mor{b, a} and Mor{a, a} = Mor(a, a)). Also, write for the subset of Mor(a, b) (resp., Mor{a, b}) consisting of non-invertible morphisms. An object a in C is said to be minimal if it is not an initial object and any monomorphism b ֒→ a in C, where b is not an initial object, is necessarily an isomorphism.
Next, we denote by C op the opposite category of C. Also, given each functor F : C → D, we shall write F op : C op → D op for the functor between the respective opposite categories naturally induced by F . We denote by C ∼ = the category whose objects are the elements of Ob(C) and whose morphisms are the isomorphisms in Mor(C). Given two categories C, D, we shall write C ∼ = D (resp., C isom ∼ = D) if C is equivalent (resp., isomorphic) to D. Recall that a category C is said to be a groupoid if the morphisms in C are all invertible. By a groupoid in C, we mean a subcategory D of C forming a groupoid such that, if a, a ′ are objects in D and u : a ∼ → a is an isomorphism in C, then u lies in Mor(D). A category C is said to be a preorder if Mor(a, b) has at most one element for all objects a, b ∈ Ob(C). We write a < b if Mor(a, b) is nonnempty; the binary relation "<" in Ob(C) is reflexive and transitive. If in addition "<" is symmetric, C is said to be an order. Note that "<" is symmetric precisely when the only isomorphisms of the preorder C are the identity morphisms. If (T, <) is a partial order, then we write T ⊥ for the order defined in such a way that Ob(T ⊥ ) = T and Mor(a, b) = ∅ precisely when a < b.
Throughout the present paper, we shall fix a universe U . Denote by Set the category consisting of U -small sets and maps between them. Denote by Cat the category consisting of U -small categories and functors between them. Moreover, denote by Ord the full subcategory of Cat consisting of U -small orders. To each U -small set T , we associate the category Dis(T ), whose objects are the elements in T and whose only morphisms are identity morphisms. The assignment T → Dis(T ) defines a functor Dis : Set → Ord.
2.2. Structure species. Let us recall the categorical reformulation of Bourbaki's structure species discussed in [ABla]. Fix a U -small category X.
(i) Let us consider a pair consisting of two functors E : X → Ord, S : X ∼ = → Set. We say that Σ is a (covariant) structure species on X if the composite Dis • S is a subfunctor of E • J, meaning that, for every a ∈ Ob(X ∼ = ), the category (Dis • S)(a) is a subcategory of (E • J)(a) and the inclusion (Dis • S)(a) ֒→ (E • J)(a) is functorial with respect to a.
(ii) Let Σ := (E, S) and Σ ′ := (E ′ , S ′ ) be structure species on X. A morphism of structure species from Σ to Σ ′ is defines as a natural transformation φ : S → S ′ such that, for every morphism u : a → b in X and for every U ∈ Ob(S(a)) and Remark 2.2.2. Let Σ := (E, S) be a structure species on X. As mentioned in the Remark following [ABla], § 2, Definition 2.1, E and S respectively correspond to the echelon and structure scheme of Σ in the traditional terminology of Bourbaki (cf. [Bou], Chap. IV, § 1).
Example 2.2.3 (Continuous maps on topological spaces). For each U -small set T , we shall write P (T ) for the power set (i.e., the set of subsets) of T . Let P + (resp., P − ) be the functor Set → Set (resp., Set op → Set) defined as follows: • For each object T in Set, we set P + (T ) := P (T ) (resp., P − (T ) := P (T )).
• For each morphism f : T → T ′ in Set, we set P + (f ) (resp., P − (f )) to be the map P + (T ) → P + (T ′ ) given by U → f (U) for every U ∈ P + (T ) (resp., the map P − (T ′ ) → P − (T ) given by U ′ → f −1 (U ′ ) for every U ′ ∈ P (T ′ )). The maps P + (f ) and P − (f ) defined for any morphism f : T → T ′ in Set are non-decreasing when both P (T ) (= P + (T ) = P − (T )) and P (T ′ ) (= P + (T ′ ) = P − (T ′ )) are equipped with the order structures determined by "⊆". Hence, the assignments T → (P + (T ), ⊆) ⊥ and T → (P − (T ), ⊆) ⊥ respectively induce functors P + : Set → Ord and P − : Set op → Ord. Now, let Z be a U -small subcategory of Set op . We write E := P + •P − | Z : Z → Set → Ord. We shall denote by the category consisting of structure species on X and morphisms between them. If there exists an equivalence of categories X ∼ → X ′ (where X ′ is another U -small category), then we can construct an equivalence of categories Sp X ∼ → Sp X ′ in an evident manner.
(i) Let F : Y → X be a functor between U -small categories. We say that F is a constructive functor on X (or simply, F is constructive) if it satisfies the following conditions: -F is faithful.
-For every a ∈ Ob(Y ) and for every isomorphism u in X with domain F (a), there exists uniquely an isomorphism u Y in Y with domain a satisfying the equality F (u Y ) = u. We often refer to the second condition as the lifting property on F .
We shall denote by Con X (6) the category consisting of constructive functors on X and morphisms between them. One may verify that Con X admits finite coproducts and fiber products. If f : X ′ → X is a functor between U -small categories, then the assignment Y → f * (Y ) := Y × X X ′ defines a functor f * : Con X → Con X ′ .
Here, we shall construct a constructive functor associated to a structure species. Let Σ := (E, S) be a structure species on X. Denote by Y Σ the category defined as follows: • The objects of Y Σ are pairs (a, T ) such that a ∈ Ob(X) and T ∈ S(a).
where "<" is the relation defined on the order E(a ′ ). The composite law for morphisms in Y Σ is defined in a natural manner. Moreover, we set to be the functor given by (a, T ) → a and u → u. This functor forms a constructive functor on X (cf. [ABla], § 2, Propositions 2.1 and 2.2). To a morphism of structure species φ : Σ → Σ ′ , we can associate a morphism of constructive functors F φ : F Σ → F Σ ′ (but we will omit the details of this construction). According to [ABla], § 4, Theorem 2, the assignments Σ → F Σ and φ → F φ together define an equivalence of categories 2.4. Property of a constructive functor. In this section, we observe one property of a constructive functor (cf. Proposition 2.4.3). Before doing so, let us recall the connectedness of a category.
Definition 2.4.1. A category C is said to be connected if it is nonempty and for any pair of objects a, b ∈ Ob(C), there exists a finite sequence (c 1 , · · · , c n ) of objects in C such that c 1 = a, c n = b, and Mor{c j , c j+1 } = ∅ for any j = 1, · · · , n − 1.
Example 2.4.2. A category C is connected if it has either an initial object or a terminal object. In particular, a category consisting of U -small sets containing either the empty set or a singleton is connected.
Next, we prove the following assertion, which will be used in the subsequent discussion.
. This shows the surjectivity of Ob(Φ). The surjectivity of Mor(Φ) can be proved by using a similar argument, so we will finish the proof here.
2.5. Constructive functor on a groupoid. This subsection deals with a specific constructive functor on a groupoid. From Proposition 2.5.2 described later, this constructive functor can be thought of as the universal covering of a topological space or a graph (cf. Example 2.5.1, Remark 2.5.3).
Let G be a U -small groupoid and e an object in G. Suppose further that G is connected, which implies Mor(a, a ′ ) = ∅ for any a, a ′ ∈ Ob(G). We shall set Y G,e to be the category defined as follows: • The objects in Y G,e are pairs (a, u) consisting of a ∈ Ob(G) and u ∈ Mor(e, a).
The composition law for morphisms in Y G,e is defined in an evident manner. One may verify that Y G,e is a connected groupoid and, for any a, a ′ ∈ Ob(Y G,e ), the set Mor(a, a ′ ) has exactly one element. The assignments (a, u) → a and v → v together define a functor which is verified to form a constructive functor on G.
Example 2.5.1. Let us consider the case where Ob(G) has exactly one element, which we denote by ⊛. Denote by Aut(⊛) the automorphism group of ⊛. Then, the category Y G,⊛ may be regarded as the Cayley graph Cay(Aut(⊛)) of the group Aut(⊛) by taking the connection set as Aut(⊛) itself, where the vertices and arcs in Cay(Aut(⊛)) are respectively associated with the objects and morphisms in Y G,⊛ .
We shall prove the following proposition concerning the constructive functor F G,e .
Proof. First, we prove the surjectivity of ev F,e in assertion (i). Let e Y be an element of F −1 (e).
In what follows, we shall construct a morphism of constructive functors F G,e → F that is mapped to e Y via ev F,e . Let us take an arbitrary object (a, u) of Y G,e . Since F is constructive, there exists a unique pair (a Y , u Y ) consisting of an object a Y in Y and a morphism u Y : and this morphism satisfies ev F,e (Φ e Y ) = e Y by construction. This implies the surjectivity of ev F,e .
Next, we prove the injectivity of ev F,e . Let Φ be a morphism F G,e → F and write e Y := ev F,e (Φ). The problem reduces to one of proving the equality Φ = Φ e Y . To this end, let us take an arbitrary element (a, u) of Y G,e . Since u defines a morphism (e, id e ) → (a, u) in Y G,e , the image Φ(u) defines a morphism e Y (= Φ((e, id e ))) → Φ((a, u)). This morphism is a unique lifting of u ∈ Mor(G) with domain e Y , so we have Φ((a, u)) = Φ e Y ((a, u)). Thus, we obtain the equality Φ = Φ e Y , as desired. This completes the proof of the bijectivity of ev F,e . Now, we prove assertion (ii). From Proposition 2.4.3 and the fact that Y G,e is a connected groupoid, it suffices to prove the injectivity of Ob(Φ) and of Mor(Φ). Suppose that there exists This completes the proof of the injectivity of Ob(Φ). A similar argument can be used to proved the injectivity of Mor(Φ).
forms a constructive functor on G. Then, it follows from assertion (ii) that the morphism Φ| G 0 : F | G 0 → F G,e obtained by restricting Φ is an isomorphism. Here, suppose that there exists an object a ∈ Ob(Y ) \ Ob(G 0 ). Denote by G 1 the groupoid in Y containing a (hence Ob(G 0 ) ∩ Ob(G 1 ) = ∅). For the same reason as above, the restriction Φ| G 1 : F | G 1 → F G,e of Φ to G 1 forms an isomorphism of constructive functors. Thus, we obtain two distinct morphisms (Φ| G 0 ) −1 , (Φ| G 1 ) −1 belonging to Mor(F G,e , F ) that coincide with each other after composing with Φ. This contradicts the assumption that Φ is a monomorphism. This implies Ob(G 0 ) = Ob(Y ). Next, suppose that there exists a morphism u ∈ Mor(Y ) \ Mor(G 0 ). Then, the two morphisms id Y and (Φ| G 0 ) −1 •Φ are distinct but identical to each other when composed with Φ. This is a contradiction, so we have Mor(Y ) = Mor(G 0 ). This implies that Y = G 0 and Φ = Φ| G 0 . Thus, Φ turns out to be an isomorphism.
Let e ′ be an object in G and w : e → e ′ a morphism in G. Then, we obtain a morphism of constructive functors To be precise, this morphism is obtained by assigning (a, u) → (a, u • w −1 ) (for each (a, u) ∈ Ob(Y G,e )) and v → v (for each v ∈ Mor(Y G,e )). From Proposition 2.5.2, (ii), Φ w turns out to be an isomorphism. The existence of Φ w implies that the isomorphism class of the constructive functor F G,e does not depend on the choice of e. By considering the case of e = e ′ and applying again Proposition 2.5.2, (ii), we see that the assignment w → Φ w defines an isomorphism of groups φ e := ev −1 F G,e ,e : F −1 G,e (e) = Aut(e) → Aut Con G (F G,e ), where Aut Con G (F G,e ) denotes the automorphism group of F G,e in Con G .
Remark 2.5.3. One may verify that Con G forms a Galois category (cf. [Gro1], Exposé V, Théorèm 4.1 and Définition 5.1) by setting ev F,e as the fiber functor. The above discussion shows that the resulting fundamental group π 1 (Con G ; ev F,e ) of Con G with base point ev F,e is isomorphic to the group Aut(e). Hence, the equivalence class of the groupoid G can be reconstructed from the fundamental group π 1 (Con G ; ev F,e ). This fact is reminiscent of "Grothendieck conjecture"-type theorems in anabelian geometry (cf. Remark 3.4.2).
Remark 2.5.4. The cultural model, as well as the paradigm for knowledge and practice, in the modern Western world envisioned by the structuralism of Lévi-Strauss and others is characterized by an organizational structure of an arborescence system that looks for the origin of "things" and for the culmination or conclusion of those "things". In this model, a small idea, like a seed, takes root and grows into a tree with a sturdy trunk supporting numerous branches, all linked to and traceable back to the original idea. ‡ The results of Proposition 2.5.2 together with the viewpoint of the previous Remark suggest that, in accordance with the ‡ Due to criticism of how an object can be explained only with such a picture, a perspective that focuses (not only on the origin and conclusion but) on the process of its creation and the possibility of its change arose later. It led to the establishment of the position known today as post-structuralism. In this context, G. Deleuze and F. Guattari used the term "rhizome" to describe a process of existence and growth that does not come from a single central point of origin (cf. [DeGu]). picture of Lévi-Strauss' structuralism, F G,e plays the role of a seed or a trunk in the system of structure species on G.

Proof of the main theorem
This section is devoted to proving Theorem A. The nontrivial portion of that theorem is the implication (b) ⇒ (a), which asserts that the equivalence class of a category X may be characterized uniquely, up to a certain indeterminacy, from the categorical structure of Sp X , or equivalently, of Con X (cf. (9)). In the following discussion, we will often speak of various things concerning Con X as being "reconstructed (or characterized) category-theoretically". By this, we mean that they are preserved by an arbitrary equivalence of categories Con X ∼ → Con X ′ (where X ′ is another U -small category). For instance, the set of monomorphisms in Con X may be characterized category-theoretically as the morphisms Φ : F → F ′ such that, for any F ′′ , the map of sets Map(F ′′ , F ) → Map(F ′′ , F ′ ) obtained by composing with Φ is injective. To simplify our notation, however, we will omit explicit mention of this equivalence Con X ∼ → Con X ′ , of X ′ , and of the various "primed" objects and morphisms corresponding to the original objects and morphisms, respectively, in Con X .
Our tactic for completing the proof of the implication (b) ⇒ (a) (i.e., recognizing the structure of X) is, as in [Moc1], [Moc2], and [Wak], to reconstruct step-by-step various partial information about X from the categorical structure of Con X .
3.1. Reconstruction of the objects and their automorphisms. The first step of the proof is to reconstruct the set of isomorphism classes of objects in Con X and their automorphism groups. Let us fix a U -small category X. Also, let us fix a skeleton X of X (i.e., a full subcategory X of X such that the inclusion X ֒→ X is an equivalence of categories and no two distinct objects in X are isomorphic).
If G and G ′ are distinct connected groupoids in X, then we see that Ob(G) ∩ Ob(G ′ ) = ∅. Moreover, for an object e of X, there exists a unique connected groupoid G e in X containing e. Thus, the assignment e → G e defines a bijective correspondence between Ob(X) and the set of connected groupoids in X. Also, we have a decomposition where the disjoint union in the right-hand side runs over the set of connected groupoids in X. Now, let e be an object in X. Denote by G the connected groupoid in X containing e (hence Ob(X) ∩ Ob(G) = {e}) and denote the natural inclusion by ι G : G ֒→ X. The assignment F → ι G * (F ) := ι G • F defines a fully faithful functor ι G * : Con G → Con X .
The following proposition can be proved immediately from the definitions of a constructive functor and ι G * (so we will omit their proofs.) Proposition 3.1.1.
(i) This functor is left adjoint to the functor ι * G (cf. (7)), which means that for F ∈ Ob(Con X ) and F ′ ∈ Ob(Con G ), there exists a functorial bijection (ii) Let F be a constructive functor on X. Suppose that there exist a constructive functor F ′ on G and a morphism F → ι G * (F ′ ). Then, there exists a constructive functor F ′′ on G with ι G * (F ′′ ) ∼ = F .
The following assertion is a direct consequence of assertion (ii) above.
Corollary 3.1.2. Let F be a constructive functor on G. Then, F is minimal in Con G if and only if ι G * (F ) is minimal in Con X .
We shall write When there is no fear of confusion, we will write F + G := F + G,e for simplicity. (This abbreviation of the notation can be justified because of the existence of (12) and the fully faithfulness of ι G * . In fact, these facts show that the isomorphism class of F + G,e depends only on G; i.e., it does not depend on the choice of the skeleton X.) Denote by † Ob(X) ⊆ Ob(Con X ) (18) the set of isomorphism classes of objects in Con X of the form F + G for some connected groupoid G in X. If G and G ′ are connected groupoids in X, then F + G ∼ = F + G ′ precisely when G = G ′ . This implies that the assignment from each element e ′ ∈ Ob(X) to the constructive functor F + G ′ , where G ′ denotes the connected groupoid in X containing e ′ , defines a bijection of sets ξ X : Ob(X) Proposition 3.1.3. Let F be a constructive functor on X. Then, F is minimal in Con X if and only if F is isomorphic to F + G for some connected groupoid G in X. In particular, the subset † Ob(X) of Ob(Con X ) can be reconstructed category-theoretically from the data Con X (i.e., of a category).
Proof. The "if" part of the required equivalence follows immediately from Proposition 2.5.2, (iii), and Corollary 3.1.2. Hence, in what follows, we shall prove the "only if" part. Let F : Y → X be a minimal object in Con X . Since this object is nonempty, there exists an object a in Y . Then, by the bijection ev F,F (a) in Proposition 2.5.2, (i), together with the adjunction relation (16), we can find a morphism of constructive functors Φ : F + G → F . It follows from Proposition 2.4.3 that the maps Ob(Φ) : Ob(Y G,e ) → Ob(Y ), Mor(Φ) : Mor(Y G,e ) → Ob(Y ) induced by Φ are surjective. Next, suppose that we are given two objects (a, u), (a, u ′ ) in Y G,e with Φ((a, u)) = Φ((a, u ′ )). After possibly composing Φ with a suitable automorphism of F + G , we may assume that a = e. Denote by v the unique endomorphism of e in G satisfying {v} = Mor((e, u), (e, u ′ )). Then, from the bijectivities of ev F,e and ev F G,e , it can immediately be seen that v = id e , i.e., (e, u) = (e, u ′ ). This implies the injectivity of Ob(Φ). Moreover, since Mor((a, u), (b, v)) is a singleton for any objects (a, u), (b, v) in Y G,e , the injectivity of Ob(Φ) implies that of Mor(Φ). Consequently, Φ turns out to be an isomorphism. This completes the proof of the "only if" part.
Let e and G be as above. Then, for each w ∈ Aut(e), the automorphism Φ w (cf. (12)) defines, via ι G * , an automorphism Φ + u of F + G . By the bijectivity of φ e (cf. (13)) and the fully faithfulness of ι G * , the assignment w → Φ + w defines an isomorphism of groups ξ where we set † Aut(e) := Aut Con X (F + G ). The following assertion can be verified immediately.
Proposition 3.1.4. The subset † Aut(e) of Mor(Con X ) together with its group structure can be reconstructed category-theoretically from the data (Con X , F + G ) (i.e., of a category and a minimal object in this category).

3.2.
Reconstruction of the non-invertible morphisms. Let us take two distinct objects e, e ′ in X. Denote by G and G ′ the connected groupoids in X containing e and e ′ respectively. Given a non-invertible morphism v ∈ Mor(e, e ′ ) (i.e., v ∈ Mor(e, e ′ ) ≇ ), we shall denote by the category determined as follows: • Mor((a, u), (a ′ , u ′ )), where Mor((a, u), (a ′ , u ′ )) : The composition law for morphisms in Y G,G ′ ,v is defined in a natural manner. Moreover, denote by the functor given by (a, u) → a (for each (a, u) ∈ Ob(Y G,G ′ ,v )) and w → w for each w ∈ Mor(Y G,G ′ ,v ). Then, F + G,G ′ ,v forms a constructive functor on X, and the natural inclusions When there is no fear of confusion, we will write F + v := F + G,G ′ ,v and Υ v := Υ G,G ′ ,v . Next, denote by C e,e ′ (24) the category defined as follows: • The objects in C e,e ′ are pairs (F, Υ) consisting of a constructive functor F : Y → X on X and a non-invertible monomorphism Υ : F + G,e ⊔ F + G ′ ,e ′ ֒→ F in Con X . • The morphisms from (F, Υ) to (F ′ , Υ ′ ) are morphisms of constructive functors Ψ : For each non-invertible morphism v : e → e ′ in X, the pair (F + G,G ′ ,v , Υ G,G ′ ,v ) introduced above specifies an object of C e,e ′ . The assignment (F, Υ) → F defines a functor C e,e ′ → Con X .
By taking account of Proposition 3.1.3, we see that the category C e,e ′ together with the functor (25) can be reconstructed category-theoretically from the data (Con X , F + G,e , F + G ′ ,e ′ ) (i.e., of a category and two distinct minimal objects in this category). We identify C e,e ′ with C e ′ ,e via the equivalence of categories C e,e ′ ∼ → C e ′ ,e obtained by switching the factors F + G,e ⊔ F + Mor(e ′ , e) ≇ ) may be regarded as elements of Ob(C e,e ′ ) and Mor(C e,e ′ ) respectively. Moreover, we denote by † Mor{e, e ′ } ≇ (⊆ Mor(Con X )) (26) the set of morphisms in Con X of the form Υ Proposition 3.2.1. Let (F, Υ) be an object in C e,e ′ . Then, the following two conditions are equivalent to each other: (a) (F, Υ) is minimal in C e,e ′ and there is no triple of morphisms in Con X such that Φ is an isomorphism and satisfies the equality In particular, the subset † Mor{e, e ′ } ≇ of Mor(Con X ) can be reconstructed category-theoretically from the data (Con X , F + G,e , F + G ′ ,e ′ ) (i.e., of a category and two distinct minimal objects in this category).
Proof. Since the implication (b) ⇒ (a) can be immediately verified from the definition of (F + v , Υ v ), we only consider the inverse direction. Let (F : Y → X, Υ) be an object in C e,e ′ satisfying the conditions in (a). Denote by Y G,e and Y G ′ ,e ′ the subcategories in Y defined as the images via Υ of Y G,e and Y G ′ ,e ′ respectively. By the lifting property on F , Y G,e and Y G ′ ,e ′ respectively specify connected groupoids in Y . Now, suppose that there exists an object a 0 in Ob(Y )\(Ob(Y G,e )⊔Ob(Y G ′ ,e ′ )). If G 0 denotes the connected groupoid in Y containing a 0 , then the minimality of (F, Υ) implies that Y must be equal to the disjoint union Y G,e ⊔ Y G ′ ,e ′ ⊔ G 0 . But this is a contradiction because of the second condition in (a). Hence, we have Ob(Y ) = Ob(Y G,e ) ⊔ Ob(Y G ′ ,e ′ ). Since Υ is a monomorphism, the functors Y G,e → Y G,e , Y G ′ ,e ′ → Y G ′ ,e ′ obtained by restricting Υ are an isomorphism (cf. the proof of Proposition 3.1.3); by using these isomorphisms, we consider Y G,e ⊔ Y G ′ ,e ′ to be a subcategory of Y . From the second condition in (a), there exists objects a ∈ Ob(Y G,e ), a ′ ∈ Ob(Y G ′ ,e ′ ) such that Mor{a, a ′ } = ∅. Hence, the subcategory Y G,e ⊔ Y G ′ ,e ′ of Y extends to a subcategory of the form Y G,G ′ ,v for some v ∈ Mor{e, e ′ }. It follows from the minimality of (F, Υ) that the inclusion Y G,G ′ ,v ֒→ Y must be an isomorphism. This completes the proof of the implication (a) ⇒ (b).

3.3.
Reconstruction of the composition law. Let e, e ′ , G, and G ′ be as above. We shall set Com(e, e ′ ) We will prove the following assertion.
Proposition 3.3.1. Let us choose u ∈ Aut(e) and v 1 , v 2 ∈ Mor{e, e ′ } ≇ . Then, the following two conditions are equivalent to each other: In particular the image of Com(e, e ′ ) can be reconstructed category-theoretically from the data (Con X , F + G,e , F + G ′ ,e ′ ), i.e., of a category and two distinct minimal objects in this category (cf. Proposition 3.1.4 for the categorytheoretic reconstruction of † Aut(−)).
Proof. The assertion follows from the various definitions involved.  3.1.4). Let G, G ′ , G ′′ , e, e ′ , and e ′′ be as above. Moreover, we will write Com(e, e ′ , e ′′ ) :
That is to say, Com(e, e ′ , e ′′ ) and Com(e, e ′ , e ′′ ) op are defined as the graphs of the composition law defining X.

Proof of Theorem A.
We can now prove Theorem A by applying the results obtained so far. First, let us prove the implication (a) ⇒ (b). To this end, it suffices to show that Sp X ∼ = Sp X op , or equivalently Con X ∼ = Con X op (cf. (9)). But this fact can be verified because the assignment from each constructive functor F : Y → X to F op : Y op → X op yields an equivalence of categories δ X : Con X ∼ → Con X op . Next, let us consider the inverse direction (b) ⇒ (a). Suppose that there exists an equivalence of categories Sp X ∼ → Sp X ′ ; this equivalence together with (9) determines an equivalence of categories α : Con X ∼ → Con X ′ . Let us fix skeletons of X and X ′ , which we denote by X and X ′ respectively. By Proposition 3.1.3, α induces, via ξ X and ξ X ′ , a bijection It follows from Propositions 3.1.4 and 3.2.1 that, for each e, e ′ ∈ Ob(X), the equivalence α induces, via ξ e,e ′ and ξ α(e),α(e ′ ) , a bijection α e,e ′ : Mor{e, e ′ } ∼ → Mor{α(e), α(e ′ )}.
Remark 3.4.1. The above proof shows that, in order to reach the conclusion, it suffices to assume the connectedness only for either X or X ′ . In other words, the connectedness of X can be characterized category-theoretically by the category Sp X .
Remark 3.4.2. The reconstruction carried out in Theorem A is reminiscent of "Grothendieck conjecture"-type theorems in anabelian geometry. Anabelian geometry is, roughly speaking, an area of arithmetic geometry that discusses the issue of how much information concerning the geometry of certain arithmetic varieties (e.g., hyperbolic curves over a number field or a padic local field) is contained in the knowledge of theétale fundamental groups, or equivalently, the categories of finiteétale coverings. The classical point of view of anabelian geometry centers around a comparison between two arithmetic varieties (or more generally, two geometric objects of the same kind) via theirétale fundamental groups and it is referred to as bi-anabelian geometry. On the other hand, mono-anabelian geometry, being an alternative and relatively new formulation, centers around the task of establishing a group-theoretic algorithm whose input data consists of a single abstract topological group isomorphic to theétale fundamental group of a single geometric object. In particular, it requires us to reconstruct, unlike the bianabelian formulation, the desired data without any mention of some "fixed reference model" copy of initial objects. For basic references, we refer the reader to [Hos], [Moc3].
As explicitly verbalized in [Son], § 3, constructive functors on a category X may be regarded, in some sense, as categorical realizations of coverings over X. In fact, if X is a groupoid G, then the results of Proposition 2.5.2 enable us to consider the constructive functor F G,e : Y G,e → G as if it were the universal covering of G. Accordingly, we might say that Sp X ( ∼ = Con X by (9)) is like the Galois category consisting of finiteétale coverings. Notwithstanding the fact that Theorem A is stated in a completely bi-anabelian way, the proof of the theorem actually furnishes a mono-anabelian algorithm that reconstructs X from Sp X . Here, however, we will omit the details of the formulation, as well as the proof, in that way.
Remark 3.4.3. C. Lévi-Strauss developed a structuralist theory of mythology which attempted to explain how seemingly fantastical and arbitrary tales could be so similar across cultures (cf. [Lev1], [Lev2]). Because he believed that there was no one authentic version of a myth, rather that they were all manifestations of the same language, he sought to find the fundamental units of myth, namely, the mytheme. The canonical formula of mythical transformation is an expression proposed in 1955 by Lévi-Strauss in order to account for the abstract relations occurring between characters and their attributes in a myth understood as the collection of its variants. According to the canonical formula, a myth is reducible to an expression: where each of these four arguments consist of a term variable (a and b), and a function variable (x and y). This formula describes a structural relationship between a set of narrative terms and their transmutative relationships. See [Mar] for a reference on this formula.
Note that J. Morava tried to develop a truly mathematical argument for the canonical formula (cf. [Mor1], [Mor2]). He proposed to interpret it as the existence of an anti-isomorphism of the quaternion group. On the other hand, as a mathematical study of myths in another direction, we might expect that some kind of symmetry or structure on the whole category of structure species satisfying the condition expressed by (43) serves as a metaphorical explanation of some truth that Lévi-Strauss expected from myths. However, at the time of writing the present paper, the author does not have any effective ideas for the development of this argument.

Appendix: Category of categories over a category
In this Appendix, we establish, as an analogy of Theorem A, the reconstruction of a category X from the category of categories over X, i.e., the category of functors F : Y → X (that are not necessarily a structure species). The conclusion differs from the case of Theorem A in that we can reconstruct (up to a certain indeterminacy) not only the equivalence class of X but also its isomorphism class. The following Proposition 4.0.1 (resp., 4.0.2; resp., 4.0.3) will be proved by using a much simpler argument than was used in the corresponding previous assertion, i.e., Proposition 3.1.3 (resp., 3.2.1; resp., 3.3.2). So we will leave their proofs to the reader.
Let X be a U -small category. Denote by the category defined as follows: • The objects are functors of the form F : The identity functor id X : X ∼ → X is a terminal object in this category. For each object e in X, we shall write Y e := Dis({e}) and write F e : Y e ֒→ X (45) for the natural functor; this functor species an object in Cat X . We denote by ‡ Ob(X) (⊆ Ob(Cat X )) (46) the set of objects in Cat X of the form F e for some e ∈ Ob(X). The assignment e → F e gives a bijection of sets Proposition 4.0.1. Let F : Y → X be an object in Cat X . Then, F is minimal in Cat X if and only if F is isomorphic to F e for some e ∈ Ob(X). In particular, the subset ‡ Ob(X) of Ob(Cat X ) can be reconstructed category-theoretically from the data Cat X (i.e., of a category).
Next, let e and e ′ be (possibly the same) objects in X. For each morphism v : e → e ′ in X, we shall set Y v to be the subcategory of X satisfying Ob(Y v ) = {e, e ′ } and Mor(Y v ) = {id e , id e ′ , v}. Denote by the natural inclusion by The inclusions Y e ֒→ Y v and Y e ′ ֒→ Y v induce the morphism in Cat X .
We shall denote by the category defined as follows: • The objects in D e,e ′ are pair (F, Υ) consisting of an object F : Y → X in Cat X and a non-invertible monomorphism Υ : F e ⊔ F e ′ ֒→ F .
It follows from Proposition 4.0.1 that the category D e,e ′ together with the functor (51) can be reconstructed category-theoretically from the data (Cat X , F e , F e ′ ) (i.e., of a category and two minimal objects in this category). Also, as in the case of "C e,e ′ " introduced in § 3.2, there exists a natural identification D e,e ′ = D e ′ ,e , by which we will not distinguish between D e,e ′ and D e ′ ,e . In particular, this identification allows us to consider (F w , Υ w )'s for various w ∈ Mor(e ′ , e) as objects in D e,e ′ . We denote by ‡ Mor{e, e ′ } (52) the set of morphisms in Cat X of the form Υ v for some v ∈ Mor{e, e ′ }. The assignment v → Υ v defines a bijection of sets ζ e,e ′ : Mor{e, e ′ } ∼ → ‡ Mor{e, e ′ }.
Proposition 4.0.2. Let (F, Υ) be an object in Cat X . The following two conditions are equivalent to each other: • (F, Υ) is minimal in D e,e ′ and there is no triple of morphisms in Cat X such that Ψ is an isomorphism and satisfies the equality Υ = Ψ • (Υ 0 ⊔ Υ ′ 0 ). • (F, Υ) is isomorphic to (F v , Υ v ) for some v ∈ Mor{e, e ′ }. In particular, the subset ‡ Mor{e, e ′ } of Mor(Cat X ) can be reconstructed category-theoretically from the data (Cat X , F e , F e ′ ) (i.e., of a category and two minimal objects in this category).
The previous three propositions enable us to prove the following assertion. (The proof is entirely similar to the proof of Theorem A.) Theorem 4.0.4. Let X and X be connected U -small categories. Then, the following conditions are equivalent to each other: Proof. The implication (a) ⇒ (b) follows immediately from the existence of an equivalence of categories Cat X ∼ → Cat X op obtained by assigning F → F op for each F ∈ Cat X . Next, we shall consider the inverse direction (b) ⇒ (a). Suppose that there exists an equivalence of categories β : Cat X ∼ → Cat X ′ . By Proposition 4.0.1, β induces, via ζ X and ζ X ′ , a bijection β : Ob(X) ∼ → Ob(X ′ ).