Some remarks on blueprints and F1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\pmb {{\mathbb {F}}}_1}$$\end{document}-schemes

Over the past two decades several different approaches to defining a geometry over F1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb F}_1}$$\end{document} have been proposed. In this paper, relying on Toën and Vaquié’s formalism (J.K-Theory 3: 437–500, 2009), we investigate a new category SchB~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {Sch}}_{\widetilde{{\mathsf B}}}$$\end{document} of schemes admitting a Zariski cover by affine schemes relative to the category of blueprints introduced by Lorscheid (Adv. Math. 229: 1804–1846, 2012). A blueprint, which may be thought of as a pair consisting of a monoid M and a relation on the semiring M⊗F1N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\otimes _{{{\mathbb F}_1}} {\mathbb N}$$\end{document}, is a monoid object in a certain symmetric monoidal category B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf B}$$\end{document}, which is shown to be complete, cocomplete, and closed. We prove that every B~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{{\mathsf B}}}$$\end{document}-scheme Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} can be associated, through adjunctions, with both a classical scheme ΣZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _{\mathbb Z}$$\end{document} and a scheme Σ̲\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{\Sigma }$$\end{document} over F1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb F}_1}$$\end{document} in the sense of Deitmar (in van der Geer G., Moonen B., Schoof R. (eds.) Progress in mathematics 239, Birkhäuser, Boston, 87–100, 2005), together with a natural transformation Λ:ΣZ→Σ̲⊗F1Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda :\Sigma _{\mathbb Z}\rightarrow \underline{\Sigma }\otimes _{{{\mathbb F}_1}}{\mathbb Z}$$\end{document}. Furthermore, as an application, we show that the category of “F1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb F}_1}$$\end{document}-schemes” defined by Connes and Consani in (Compos. Math. 146: 1383–1415, 2010) can be naturally merged with that of B~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{{\mathsf B}}}$$\end{document}-schemes to obtain a larger category, whose objects we call “F1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb F}_1}$$\end{document}-schemes with relations”.


A quick overview of F 1 -geometry
The nonexistent field F 1 made its first appearance in Jacques Tits's 1956 paper Sur les analogues algébriques des groupes semi-simples complexes [26]. 1 According to Tits, it was natural to call "n-dimensional projective space over F 1 " a set of n + 1 points, on which the symmetric group n+1 acts as the group of projective transformations. So, n+1 was thought of as the group of F 1 -points of SL n+1 , and more generally it was conjectured that, for each algebraic group G, one ought to have W (G) = G(F 1 ), where W (G) is the Weyl group of G.
A further strong motivation to seek for a geometry over F 1 was the hope, based on the multifarious analogies between number fields and function fields, to find some pathway to attack Riemann's hypothesis by mimicking André Weil's celebrated proof. The idea behind that, as explicitly stated in Yuri Manin's influential 1991-92 lectures [21] and in Kapranov and Smirnov's unpublished paper [13], was to regard Spec Z, the final object of the category of schemes, as an arithmetic curve over the "absolute point" Spec F 1 . Manin's work drew inspiration from Kurokawa's paper [14] together with Deninger's results about "representations of zeta functions as regularized infinite determinants [7][8][9] of certain 'absolute Frobenius operators' acting upon a new cohomology theory". Developing these insights, Manin  , (1.1) where the notation reg ρ and det reg refers to "zeta regularization" of infinite products and the last hypothetical equality "postulates the existence of a new cohomology theory H • ? , endowed with a canonical 'absolute Frobenius' endomorphism ". He conjectured, moreover, that the functions of the form s−ρ 2π in Eq. 1.1 could be interpreted as zeta functions according to the definition where "Tate's absolute motive" T was to be "imagined as a motive of a one-dimensional affine line over the absolute point, T 0 = • = Spec F 1 ".
The first full-fledged definition of variety over "the field with one element" was proposed by Christophe Soulé in the 1999 preprint [24]; five years later such definition was slightly modified by the same author in the paper [25]). Taking as a starting point Kapranov and Smirnov's suggestion that F 1 should have an extension F 1 n of degree n, Soulé insightfully posited that Let R be the full subcategory of the category Ring of commutative rings generated by the rings R n , n ≥ 1 and their finite tensor products. An affine variety X over F 1 is then defined as a covariant functor R → Set plus some extra data such that there exists a unique (up to isomorphism) affine variety X Z = X ⊗ F 1 Z over Z along with an immersion X → X Z satisfying a suitable universal property [25,Définition 3]. In particular, one has a natural inclusion X (F 1 n ) ⊂ (X ⊗ F 1 Z)(R n ) for each n ≥ 1. A notable result proven by Soulé was that smooth toric varieties can always be defined over F 1 .
To formalize F 1 -geometry Anton Deitmar adopted, in 2005, a different approach, which can be dubbed as "minimalistic" (using the evocative terminology introduced by Manin in [22]). In his terse paper [4], Deitmar associates to each commutative monoid M its "spectrum over F 1 " Spec M consisting of all prime ideals of M, i.e. of all submonoids P ⊂ M such that x y ∈ P implies x ∈ P or y ∈ P. The set Spec M can be endowed with a topology and with a structure (pre)sheaf O M via localization, just as in the usual case of commutative rings. A topological space X with a sheaf O X of monoids is then called a "scheme over F 1 ", if for every point x ∈ X there is an open neighborhood U ⊂ X such that (U , O X | U ) is isomorphic to (Spec M, O M ) for some monoid M. The forgetuful functor Ring → Mon has a left adjoint given by M → M ⊗ F 1 Z (in Deitmar's notation), and this functor extend to a functor -⊗ F 1 Z from the category of schemes over F 1 to the category of classical schemes over Z. Tit's 1957 conjecture stating that G L n (F 1 ) = n can be easily proven in Deitmar's theory. Indeed, since F 1 -modules are just sets and F 1 n ⊗ F 1 Z has to be isomorphic Z n , it turns out that F 1 n can be identified with the set {1, . . . , n} of n elements. Hence G L n (F 1 ) = Aut F 1 (F 1 n ) = Aut(1, . . . , n) = n .
It is not hard to show, moreover, that the functor G L n on rings over F 1 is represented by a scheme over F 1 [4,Prop. 5.2]. As for zeta functions, Deitmar defines, for a scheme X over F 1 and for a prime p, the formal power series where F p n stands for the field of p n elements with only its monoidal multiplicative structure and X (F p n ) denotes the set of F p n -valued points of X , and proves that Z X ( p, T ) coincides with the Hasse-Weil zeta function of X ⊗ F 1 F p n [4,Prop. 6.3]. Albeit elegant, this result is a bit of a letdown, for-as the author himself is ready to admit-it is clear that "this type of zeta function [...] does not give new insights".
A natural and extremely general formalism for F 1 -geometry was elaborated by Bertrand Toën and Michel Vaquié in their 2009 paper [27], tellingly entitled Au dessous de Spec Z, whose approach appears to be largely inspired by Monique Hakim's work [11]. The authors there showed how to construct an "algebraic geometry" relative to any symmetric monoidal category C = (C, ⊗, 1), which is supposed to be complete, cocomplete and to admit internal homs. The basic idea is that the category CMon C of commutative (associative and unitary) monoid objects in C can be taken as a substitute for the category of commutative rings (the monoid objects in the category Ab = Z -Mod of Abelian groups) to the end of defining a suitable notion of "scheme over C". Each object V of CMon C gives rise to the category V -Mod of V -modules and each morphism V → W in CMon C determines a change of basis functor -⊗ V W : V -Mod → W -Mod; the category of commutative V -algebras can be realized as the category of commutative monoids in V -Mod and is naturally equivalent to the category V /CMon C . An affine scheme over C is, by definition, an object of the opposite category Aff C = CMon op C and the tautological contravariant functor CMon C → Aff C is called Spec( -). By means of the pseudo-functor M that maps an object V in CMon C to the category of V -modules and a morphism Spec V → Spec W to the functor -⊗ V W : V -Mod → W -Mod, one may introduce the notions of "Zariski cover" and "flat cover" ("M-faithfully flat in Toën and Vaquié's terminology; see Definition 2.4 and Remark 2.5 below) and use such notions to equip Aff C with two distinct Grothendieck topologies, called, respectively, the flat and the Zariski topology. These topologies determine two categories of sheave on Aff C , namely Sh flat (Aff C ) ⊂ Sh Zar (Aff C ) ⊂ Presh(Aff C ). At this point, mimicking what is done in classical algebraic geometry, a "scheme over C" is defined as a sheaf in Sh Zar (Aff C ) that admits an affine Zariski cover (see Definition 2.6 and Definition 2.7 below). If we take as C the category Set of sets endowed with the monoidal structure induced by the Cartesian product, then the category Aff Set is nothing but the category Mon op and the objets of the category Sch Set can be thought of -as remarked by Toën and Vaquié -as "schemes over F 1 ". Actually, as proven by Alberto Vezzani in [28], such schemes, that we shall call monoidal schemes, turn out to be equivalent to Deitmar's schemes.
Deitmar's schemes appear therefore to constitute the very core of F 1 -geometry, not just because their definition is rooted in the basic notion of prime spectrum of a monoid, but especially because they naturally fit into the categorical framework established by Toën and Vaquié in [27], which admits generalizations in many directions (e.g. towards a derived algebraic geometry over F 1 ). Nonetheless, they are affected by some intrinsic limitations, which are clearly revealed by a result proven by Deitmar himself in 2008 [6,Thm. 4

.1]:
Theorem Let X be a connected integral F 1 -scheme of finite type. 2 Then every irreducible component of X C = X Z ⊗ Z C is a toric variety. The components of X C are mutually isomorphic as toric varieties.
Since every toric variety is the lift X C of an F 1 -scheme X , the previous theorem entails that integral F 1 -schemes of finite type are essentially the same as toric varieties. Now, semisimple algebraic groups are not toric varieties, so it is apparent that Deitmar's F 1 -schemes are too little flexible to implement Tits's conjectural program.
A possible generalization of Deitmar's geometry over F 1 was proposed by Olivier Lorscheid, who introduced the notions of "blueprint" and "blue scheme" [16]. The basic idea can be illustrated through the following example. The affine group scheme (SL 2 ) Z over the integers is defined as As the relation any naive attempt to adapt the previous definition to get a scheme over F 1 will necessarily be unsuccessful. The notion of "blueprint" just serves serves the purpose of getting rid of this difficulty: where R is a semiring and A is a multiplicative subset of R containing 0 and 1 and generating R as a semiring. A blueprint morphism f : The rationale behind this definition can be explained by considering the following situation: if one is given a monoid A and some relation which does not makes sense in A but becomes meaningful in the semiring A ⊗ F 1 N, then one can look at the blueprint In the same vein as Deitmar's approach, Lorscheid [16] associates to each blueprint B its spectrum Spec B, which turns out to be a locally blueprinted space (i.e. a topological space endowed with a sheaf of blueprints, such that all stalks have a unique maximal ideal). An affine blue scheme is then defined as a locally blueprinted space that is isomorphic to the spectrum of a blueprint, and a blue scheme as a locally blueprinted space that has a covering by affine blue schemes. Deitmar's schemes over F 1 and classical schemes over Z are recovered as special cases of this definition.

About the present paper
A natural question arises: do blue schemes fit into Toën and Vaquié's framework? This problem was addressed by Lorscheid himself in his 2017 paper [18] and answered in the negative. Nonetheless, it is possible-as already pointed out in [18]-to define a category of schemes (here called B-schemes) relative (in Toën and Vaquié's sense) to the category of blueprints. Our first aim is to study these schemes by introducing the category of blueprint in a purely functorial way, as the category of monoid objects in a closed, complete and cocomplete symmetric monoidal category B.
There is a natural adjunction ρ σ : Aff B → Aff Set * between the category of affine B-schemes and that of affine monoidal schemes. However, since the functor ρ is not continuous w.r.t. the Zariski topology, this adjunction does not give rise to a geometric morphism between the corresponding category of schemes. This hurdle may be sidestepped by introducing a larger category B containing B and by considering the category of those schemes in Sch B that admit a Zariski cover by affine B-schemes.
Such schemes, by a slight abuse of language, will be called B-schemes. It will be proved that the adjunction ρ σ above induce an adjunction ρ σ between the category of B-schemes and that of affine monoidal schemes. Moreover, it will be shown that every B-scheme generates a pair ( , Z ), where is a monoidal scheme and Z a classical scheme, together with a natural transformation : More in detail the present paper is organized as follows.
After briefly recalling in Sect. 2 the fundamental notions of "relative algebraic geometry" and fixing our notation, in Sect. 3  As proven in Theorem 3.5, the category B-which corresponds to the category of pointed set endowed with a pre-addition structure introduced in [18, §4]-carries a natural structure of symmetric monoidal category. Moreover, this structure is closed, complete, and cocomplete. So, the category B possesses all the properties necessary to carry out Toën and Vaquié's program.
It is quite straightforward to show (Proposition 3.6) that the category Blp of monoid objects in B coincides with the category of blueprints (this result was already stated, in equivalent terms, in [18, Lemma 4.1], but we provide a detailed and completely functorial proof). Thus, by applying Toën and Vaquié's formalism to the category B, we define the category Aff B = Blp op of affine B-schemes and then the category Sch B of B-schemes.
The core of our paper is Sect. 4. The natural adjunction between the category Mon 0 and the category Set * gives rise to an adjunction Aff Mon 0 | -| Aff Set * -⊗ F 1 N that factorizes as shown in the following diagram , and by defining the category Sch B of B-schemes as the subcategory of Sch B whose objects admit a Zariski cover by affine schemes in Aff B (Definition 4.15). So, a B-scheme is locally described by blueprints. In this way, one shows (Theorem 4.14) that there is a geometric morphism It follows (see Definition 4.16 and the ensuing remarks) that each B-scheme determines the following geometric data: In Sect. 5, as an application of our approach, we investigate the relationship of B-schemes and F 1 -schemes in the sense of Alain Connes and Caterina Consani [1]. According to their definition [1,Def. 4.7], an F 1 -scheme is a triple ( , Z , ), where is a monoidal scheme, Z is a scheme over Z, and is natural transformation → Z •( -⊗ F 1 Z), such that the induced natural transformation •| -| → Z , when evaluated on fields, gives isomorphisms (of sets). Thus, the category of B-schemes and that of F 1 -schemes can be combined into a larger category, namely their fibered product over the category of monoidal schemes, whose objects will be called F 1 -schemes with relations (Definition 5.3). In more explicit terms, a B-scheme determining the pair ( , Z ) and an F 1 -scheme ( , Z , ) will give rise to a F 1 -scheme with relations denoted by the quadruple ( , Z , Z , ). The main motivation behind this notion is to combine in a single geometric object both the advantages of blueprint approach and the benefits of Connnes and Consani's definition (cf. Remark 5.4 for a better explanation). Each F 1 -scheme with relations ( , Z , Z , ) (with a slight modification of our terminology, see Convention 5.5) determines a natural transformation 1 : Z → Z and a natural transformation where B is a certain pullback sheaf on the category Ring (defined by the diagram 5.4). This implies that, given a B-scheme underlying a F 1 -scheme with relations, we can think of its "F 1 q−1 -points" in two different senses, and therefore count them in two different ways, as stated in Proposition 5.6 and in Theorem 5.7. An interesting case is when the F 1 n -points of the underlying monoidal scheme are counted by a polynomial in n. Theorem 4.10 of [1] shows that, if ( , Z , ) is an F 1 -scheme such that the monoidal scheme is noetherian and torsion-free, then # ( For an F 1 -scheme with relations ( , Z , Z , ) such that the underlying B-scheme is noetherian and torsion-free (Definition 5.11), we introduce the polynomial and prove (Proposition 5.14) that Q( , n) ≤ P( , n). Finally, we would like to emphasize that our approach to blueprints, being entirely functorial, seems to be appropriate to carry out a "derived version" of the category of B-schemes. In fact, in quite general terms, a definition of "derived B-scheme" could be obtained by replacing, in our definition of B-scheme, the category Set (resp. Set * ) by the category S of spaces (resp. S * of pointed spaces) and the notion of monoid object by that of E ∞ -algebra. This issue will be the object of future work.

Schemes over a monoidal category
For the reader's convenience, we start by giving a quick résumé of some of the basic constructions of the "relative algebraic geometry" developed in [27, §2].
Let C = (C, ⊗, 1) be a symmetric monoidal category (1 is the unit object), and denote by CMon C the category of commutative (associative and unitary) monoid objects in C.
We assume that C is complete, cocomplete, and closed (i.e., for every pair of objects X , Y , the contravariant functor Hom C ( -⊗ X , Y ) is represented by an "internal hom" set Hom(X , Y )).
The assumptions on C imply, in particular, that the forgetful functor which maps an object X to the free commutative monoid object L(X ) generated by X .
For each commutative monoid V in CMon C one may introduce the notion of Vmodule (cf. [12, p. 478]). The category V -Mod of such objects has a natural symmetric monoidal structure given by the "tensor product" ⊗ V ; this structure turns out to be closed. Given a morphism V → W in CMon C , there is a change of basis functor Note that the category of commutative monoids in V -Mod -i.e. the category of commutative V -algebrasis naturally equivalent to the category V /CMon C .
The category Aff C of affine schemes over C is, by definition, the category CMon op C . Given an object V in CMon C the corresponding object in Aff C will be denoted by Spec V .
To define, in full generality, the category of schemes over C one follows the standard procedure of glueing together affine schemes. To this end, one first endows Aff C with a suitable Grothendieck topology. Let us recall the general definition. Definition 2.1 Let G be any category. A Grothendieck topology on G is the assignment to each object U of G of a collection of sets of arrows {U i → U } called coverings of U so that the following conditions are satisfied: A category with a Grothendieck topology is a called a site.

Remark 2.2
As it is clear from the definition above, a Grothendieck topology on a category G is introduced with the aim of glueing objects locally defined, and what really matters is therefore the notion of covering. So, in spite of its name, a Grothendieck topology could better be thought of as a generalization of the notion of covering rather than of the notion of topology (notice, for example, that, though the maps U i → U in a covering can be seen as a generalization of open inclusions U i ⊂ U , no condition generalizing the topological requirement about unions of open subsets is prescribed).
Given a site G and a covering U = {U i → U } i∈I , we denote by h U the presheaf represented by U and by h U ⊂ h U the subpresheaf of those maps that factorise through some element of U.
Coming back to our symmetric monoidal category C, the associated category of affine schemes Aff C can be equipped with two different Grothendieck topologies by means of the following ingenious definitions (which, of course, generalize the corresponding usual definitions in "classical" algebraic geometry). One says [27, Def. 2.9, 1), 2), 3)] that a morphism f : • an epimorphism if, for any Z in CMon C , the functor

Remark 2.5
The previous definition is actually a particular case of a more general construction. Indeed, as shown in [27], to define a topology on a complete and cocomplete category D is enough to assign a pseudo-functor M : D op → Cat satisfying the the following conditions: in D, the natural transformation q * r * ⇒ r * q * is an isomorphism.
In terms of such a functor one can define the notion of M-faithfully flat cover [27,Def. 2.3] and the associated pretopology [27,Prop. 2.4], which induces a topology on D.
In the classical theory of schemes, D is the category Ring op of affine schemes and, for each X = Spec A, M(A) is the category of quasi-coherent sheaves on X . When starting with a monoidal category C satisfying our assumptions, D is the category Aff C and the pseudo-functor M maps an object V in CMon C to the category of V -modules and a morphism Spec V → Spec W to the functor - What we have called "flat cover" correspond to Toën-Vaquié's "M-faithfully flat cover" (cf. [27, Def. 2.8, Def. 2.10]). When D is endowed with a topology, a natural question that arises is how the pseudofunctor M behaves with respect to it. It can be proven ([27, Th. 2.5] that M is a stack with respect to that topology (for the notion of a stack, the reader may consult [29]).
By making use of flat covers and Zariski covers introduced in Definition 2.4 we may equip the category Aff C with two distinct Grothendieck topologies, called, respectively, the flat and the Zariski topology. Correspondingly, there are two categories of sheaves on Aff C , namely Notice that, for each affine scheme , the presheaf Y ( ) given by the Yoneda embed- Cor. 2.11, 1)]; this sheaf will be denoted again by .
The next and final step is to define the category of schemes over the category C. We first have to introduce the notion of affine Zariski cover in the category Sh Zar (Aff C ).  Definition 2.7 A scheme over the category C is a sheaf F in Sh Zar (Aff C ) that admits an affine Zariski cover. The category of schemes over C will be denoted by Sch C .

Notation and examples
Primarily to the purpose of fixing our notational conventions, we now briefly describe the basic examples of symmetric monoidal categories we shall work with in the sequel of the present paper.
• The category Set of sets can be endowed with a monoidal product given by the Cartesian product. Then (Set, ×, * ) is a symmetric monoidal category and CMon Set = Mon is the usual category of commutative, associative and unitary monoids. • The category Set * of pointed sets can be endowed with a monoidal product given by the smash product ∧; in this case, the unit object is the pointed set S 0 consisting of two elements. Then (Set * , ∧, S 0 ) is a symmetric monoidal category and CMon Set * = Mon 0 is the category of commutative, associative and unitary monoids with "absorbent object" (such an object will be denoted by 0 in multiplicative notation and by −∞ in additive notation). • The category Mon can be endowed with a monoidal product ⊗ defined in the following way: R ⊗ R is the quotient of the product R × R by the relation ∼ such that (nr, r ) ∼ (r , nr ) for each (n, r , r ) ∈ N × R × R . Clearly, the unit object is the additive monoid (N, +). Then (Mon, ⊗, N) is a symmetric monoidal category and CMon Mon = SRing is the category of commutative, associative and unitary semirings. • The category Ab = Z -Mod of Abelian groups can be endowed with a monoidal product ⊗ Z given by the usual tensor product of Z-modules. Then (Ab, ⊗ Z , Z) is a symmetric monoidal category and CMon Ab = Ring is the category of commutative, associative and unitary rings.
For the functor L : C → CMon C defined in Eq. 2.1 as left adjoint to the forgetful functor | -| : CMon C → C we shall adopt the following special conventions: where U is the monoid consisting of just one element (the notation being motivated by the identity U ⊗ U N = N); • if C = Mon 0 , L will be denoted by where F 1 is the object of Mon 0 consisting of two element, namely F 1 = {0, 1} in multiplicative notation (also in this case, the notation is motivated by the identity All symmetric monoidal categories Set, Set * , Mon, Mon 0 , Ab described above are complete, cocomplete, and closed, so we can apply the machinery of Toën-Vaquié's theory illustrated in Subsect. 2.1 and define, for each of these categories, the corresponding category of schemes over it. In this way, when C = Ab, one unsurprisingly recovers the usual notion of classical scheme. A more intriguing example is provided by the case of C = Set.

Example 2.8 Monoidal schemes
An object of the category Sch Set is a "scheme over F 1 " in the sense of [4]. The equivalence between the two definitions was proved in [28]. We recall that, if M is a commutative monoid, its "spectrum over F 1 " Spec M can be realized as the set of prime ideals of M and given a topological space structure.
In the present paper we shall call an object in Sch Set a monoidal scheme and use the name of "F 1 -scheme" for a different kind of algebro-geometric structures (see Definition 5.1).

The category of blueprints
The notion of blueprint was introduced by Olivier Lorscheid in his 2012 paper [16].
where R is a semiring and A is a multiplicative subset of R containing 0 and 1 and generating R as a semiring. A blueprint morphism f : Notice that, given a blueprint morphism f : , its restriction f | A 1 : A 1 → A 2 is a monoid morphism that uniquely determines f on the whole of R 1 .
The idea underlying the notion of blueprint can be illustrated as follows. Some equivalence relations that do not make sense in a monoid A may be expressed in the semiring A ⊗ F 1 N. Now, any equivalence relation R on a semiring S induces a projection S → S/R and can indeed be recovered by such a map. So, the assignment of a pair (A, A ⊗ F 1 N → R) is to be interpreted as the datum of a monoid A plus the relation on A ⊗ F 1 N given by the epimorphism A ⊗ F 1 N → R. Example 3.2 Consider the monoid A T = N ∪{−∞} (in additive notation, corresponding to {T i } i∈N∪{−∞} in multiplicative notation) and the corresponding free semiring A T ⊗ F 1 N of polynomials in T with coefficient in N (the functor -⊗ F 1 N has been introduced in eq. 2.4). Notice that Spec A T has two points, namely the prime ideals {−∞} and (N \{0}) ∪{−∞}, which embed in Spec A T ⊗ F 1 N (we are loosely thinking of Spec A T ⊗ F 1 N as the underlying topological space). Now, if one takes a closed subset of Spec A T ⊗ F 1 N and intersects it with Spec A T , one could naively think that the intersection is nonempty only when the chosen closed subset is defined by some relation in A T . However, this is not the case: for instance, the relation 2T = 1, which makes the ideal (T ) trivial, cannot be expressed in the monoid A T . According to Lorscheid's idea, one can represent this affine "monoidal scheme" by considering the pair (A T , The category of blueprints can be given a handier description, which makes it easier to characterise it as the category of commutative monoids in a suitable symmetric monoidal category. Let us consider the functor -⊗ F 1 N : Mon 0 → SRing (introduced in eq. 2.4)

Definition 3.3
The category Blp is the full subcategory of -⊗ F 1 N/SRing whose objects (A, A ⊗ F 1 N → R) satisfy the conditions:  Remark 3. 4 The category B above corresponds to the category of pointed set endowed with a pre-addition structure, as described in [18, §4].

Theorem 3.5
The category B carries a natural structure of symmetric monoidal category. Moreover, this structure is closed, complete, and cocomplete.
Proof In the category B there is a natural symmetric monoidal product given by the first morphism maps n(x, x ) to nx ⊗ x and is an isomorphism (in other words, the functor N[ -] is monoidal).
Since We now show that the monoidal category B is closed. Let us define the internal hom functor by setting 4) where N M is the image of the map (the second map above is the counit of the adjunction). Let us check the adjunction property. For each map Summing up, a map as in eq. 3.5 is equivalent to a map from X to the pullback defined by the diagram along with a compatible map M → L N in such a way that the following diagram commutes: This shows that the internal hom functor in eq. 3.4 is indeed a right adjoint to the monoidal product functor in eq. 3.3.
We wish now to show that the category B is complete and cocomplete. First we prove that it admits colimits. Given a diagram whose objects are we claim that its colimit is the object because of the colimit properties in the categories Set * and Mon 0 and because the functor N[ -] preserves colimits. If two elements x, y ∈ lim − → X i have the same image m ∈ lim − → M i , then their images in the first component of C are mapped by the morphism in the second component to the same element. So, the images of x and y do coincide, just because C is an object of B. It follows that the map from the diagram in C uniquely factorises through B, so that our claim is proved.
Second we prove that B admits limits. Given a diagram as above, we claim that its limit is the object last isomorphism holds since | -| preserves limits, being a right adjoint) induced by the maps X i → |M i |; the maps from B to the diagram are the obvious ones. It is clear that B is an object of B: the surjectivity condition holds by definition, while for the injectivity condition it is enough to note that it holds when the limit is either an arbitrary product or an equalizer (see [20], Theorem V.2.1). Consider now a map from an object C to the given diagram. In the category N[ -]/Mon 0 such a map uniquely factorises through the object (lim  Similarly, the top arrow induces a semiring structure one the monoid N[X ]. In this case, since the multiplication is given by the application of the free monoid functor N[ -] to the multiplication m of X , the resulting semiring is nothing but the free semiring X ⊗ F 1 N generated by the monoid (X , m). The commutativity of the diagram ensures that the multiplication on X is consistent with that on M, so that X can still be seen as a subobject of |M|.
In conclusion, a monoid object in the category B is a blueprint, and it is also obvious that any blueprint can be obtained this way.

Remark 3.7 Theorem 3.5 and Proposition 3.6 should hopefully provide a full elucidation of [18, Lemma 4.1].
We have shown that the category of blueprints fits in with the general framework proposed by Toën and Vaquié, so we can apply the formalism of Subsection 2.1 to define the category of schemes over B.

B-schemes
This sections aims to show that the natural adjunction between the categories Aff Mon 0 and Aff Set * factorizes through an adjunction between the categories Aff Mon 0 and Aff B and an adjunction between the categories Aff Set * and Aff B , whose right adjoints induce functors between the corresponding categories of relative schemes. where The adjunctions above induce opposite adjunctions between the corresponding categories of affine schemes. We have therefore the following diagram We now wish to show that the functors in diagram 4.7 satisfy the conditions that are required to apply [ (2) the right adjointG is conservative; (3) the functorG preserves filtered colimits.
Proof (1) and (2) are straightforward. As for (3), we have to show that the right adjoint preserves filtered colimits, which is also quite obvious. The colimit of a filtered diagram (   the natural transformation between the two compositions is an isomorphism. We wish to prove that an analogous property holds when one considers a flat morphism in the category Blp. As usual, it will be enough to work in the category - such that N is a subset of |M| and generates it as a module, together with an action of A on N and an action of R on M, such that the former is the restriction of the latter. If M is an R-module M, its associated A R -module is the (R, R ⊗ F 1 N → R)-module (|M|, M), whose A R -module structure is induced by the map given by the pair of immersions ι : A → R and ι ⊗ F 1 id : where the latter fits in the commutative square The category R -Mod can therefore be identified with the full subcategory of the category of We have now to show that, for any flat morphism the natural transformation between the two compositions is an isomorphism. As for the first component, the commutativity up isomorphism of the above diagram is straightforward. As for the second component, that can be easily shown by adapting the argument in proof of Prop. 3.6 of [27]. The statement then follows from [27,Cor. 2.22].
the horizontal map sends an object (N , M) to the set N endowed with an action of the monoid A. Since tensor products are defined "componentwise", the diagram commutes.  This drawback may be sidestepped by proceeding as follows: 1) omit the requirement that the map A → |A ⊗ F 1 N| → |R| is a monomorphism in Definition 3.3 and define a category Blp that contains the category Blp of blueprints; analogously, by omitting the second condition in eq. 3.2, define a category B containing B; 2) prove that there is a functor ρ : Aff Set * → Aff B that is continuous w.r.t. the Zariski topology; 3) define the category of schemes Sch B associated to this new category; 4) restrict our attention to the subcategory of Sch B consisting of schemes that admit a cover by affine schemes in the category Aff B .

B-schemes
More precisely, the categories B and Blp are defined in the following way. We denote again by ρ : Blp → Mon 0 the forgetful functor, ρ(A, A ⊗ F 1 N → R) = A; analogously to adjunction 4.5, there is an adjunction by the equivalence relation generated by am = bm, for each m ∈ N[X i ] and for each pair (a, b) in the relation defining the quotient R.

Remark 4.9
A particular case of Lemma 4.8(b) is the following. Given an object Proof Let be any Zariski cover in the category Aff B . We have to prove that {Spec A i → Spec A} i∈I is a Zariski cover in Aff Set * . To do that, by taking into account [27, Déf. 2.10], we have to check the following four points: (1) To show that, for each i, Spec A i → Spec A is flat, that is that is exact. By applying Lemma 4.8(a) to any finite diagram, this follows from the flatness of the morphism Spec( and from the fact that ρ preserves limits, being a right adjoint. (2) To show that there is a finite subset J ⊂ I such that Analogously as above, the domain of the second component of the coequalizer C of f , g in -⊗ F 1 N/SRing is the coequalizer of Because of the universal property of colimits, there is a commutative diagram giving rise to a commutative diagram in SRing, whose rows are coequalizers and where the map C → T is the second component of the coequalizer of f , g in the category -⊗ F 1 N/SRing. As the middle vertical map and the bottom right one are surjective, so is the map C → T .  Hence, the adjunction 4.13 induces an adjunction ρ σ : Sch B → Sch Set * .
Proof We already proved that σ preserves the relevant subcategory of schemes in Proposition 4.5. So all we have to prove is that ρ preserves the relevant subcategory of schemes. In view of [27,Proposition 2.18], it suffices to observe that the following properties of ρ are satisfied: • it preserves coproducts (for it is a left adjoint), and affine schemes; • it preserves finite limits (by Proposition 4.12) and Zariski opens of affine schemes (by Lemma 4.8(b) and by the fact that ρ preserves finite limits); • it preserves images (since it preserves finite limits and colimits) and diagonal morphisms; • it preserves quotients, since it preserves colimits.

Definition 4.15
A scheme in Sch B that admits a Zariski cover by affine schemes in Aff B will be called (by a slight abuse of language) a B-scheme. The category of such schemes will be denoted by Sch B .
The rationale behind this definition is that, while B-schemes retain all good local properties of B-schemes (namely, the properties of blueprints), one gains the advantages of working in the wider and more comfortable environment of the category Sch B .
Notice that the adjunction in Theorem 4.14 obviously restrict to an adjunction A B-scheme gives rise, through the functors ρ and F Z , to a pair consisting of a monoidal scheme and a classical scheme. There is a natural transformation Z → ⊗ F 1 Z, which is obtained via the unit of the adjunction ρ σ and by applying the functor F Z . By definition, there is indeed a map where the isomorphism is given by the natural isomorphism In the affine case, such a map is simply realized as the bottom arrow of the map between arrows where the top and the left map are identities. Summing up, a B-scheme induces therefore the following objects: • a monoidal scheme ; (4.18) • a (classical) scheme Z over Z; (4.20) We shall say that the B-scheme generates the pair ( , Z ), the natural transformation 4.20 being omitted.

An application: B-schemes and F 1 -schemes
The geometric data 4.18, 4.19, 4.20 appear to be similar to (but different from) those used by A. Connes and C. Consani [1] in their definition of F 1 -scheme, which is as follows. is a natural transformation → Z • ( -⊗ F 1 Z), such that the induced natural transformation • | -| → Z , when evaluated on fields, gives isomorphisms (of sets). 3 A manifest difference between B-schemes and F 1 -schemes is, of course, the direction of the natural transformation linking the monoidal scheme and the classical scheme. Moreover, the condition on in Definition 5.1(3) may fail to be fulfilled in the case of B-schemes, as shown by the following example.

Example 5.2
Consider a pair (A, R → A ⊗ F 1 Z) defining an affine F 1 -scheme in the sense Definition 5.1. Notice that, in this case, the natural transformation calculated on a field k corresponds to mapping a prime ideal p of A ⊗ F 1 Z plus an immersion A ⊗ F 1 Z/p → k to their restrictions to R; the requirement is that this is a bijection.
On the other hand, according to the general idea underlying the notion of blueprint, if the pair (A, R) is associated with an affine B-scheme (which is, of course, the same thing as an affine B-scheme), then the ring R encodes the information of a relation R intended to reduce the number of ideals of A. Take for instance the case (A, A⊗ F 1 Z → R), with A = N ∪ { −∞} (additive notation) and R = A ⊗ F 1 Z/(2T − 1). Then, N is an ideal not coming from any ideal of R, since T is invertible (in more algebraic terms, we are saying that the map to any field k sending T to 0 can not be lifted to a map from R to k).
The category Sch B and that of F 1 -schemes may be combined into a larger category.

Definition 5.3
The category of F 1 -schemes with relations is the fibered product of the category Sch B of B-schemes and that of F 1 -schemes over the category of monoidal schemes. Thus, a B-scheme generating the pair ( , Z ) and an F 1 -scheme ( , Z , ) will determine a F 1 -scheme with relations denoted by the quadruple ( , Z , Z , ).

Remark 5.4
Recall that the aim of Lorscheid's definition of blueprint is to increase the amount of closed subschemes of a monoidal scheme. If we loosely refer to the features of the underlying topological space as "shape" of the scheme, we could say that the category of B-schemes (or that of B-schemes) adds "extra shapes" to Deitmar's category of monoidal schemes.
Consider now F 1 -schemes, and let us restrict our attention to the affine case. So, we just have a ring R, a monoid M, and a map R → M ⊗ F 1 Z. Since it is required, by definition, that points remain the same, the monoid is not enriched with "extra shapes". However, if we think of the given map as a restriction map between the spaces of functions of the affine schemes M ⊗ F 1 Z and R, we can interpret the datum of the F 1 -scheme as an enlargement of the space of functions of the affine monoidal scheme M.
In conclusion, an F 1 -scheme with relation, according to the definition 5.3, allows us both to add "extra shapes" to the underlying monoidal scheme and to enlarge its space of functions.
As an example, consider the affine F 1 -scheme with relation given by the free monoid on four generators and the data The B-scheme component on the right has been already taken into consideration in the Introduction; the F 1 -scheme component on the left adds a nilpotent component to the ring of functions.
Notice that the classical scheme Z is derived from the B-scheme via the functor F Z : Blp → Ring (Definition 4.16). This means, in particular, that the affine B-scheme = (M, M ⊗ F 1 N → R) generates the affine classical scheme Z = R ⊗ N Z. So Definition 5.3 indicates that, as long as we wish to investigate a relationship between this affine B-scheme with an F 1 -scheme and its associated affine classical scheme Z , we are no longer concerned with the "monoid relations" given the map M ⊗ F 1 N → R, but only with the "ring relations" given by the map M ⊗ F 1 Z → R ⊗ N Z (cf. eq. 4.20).
From this viewpoint it appears more natural to work with blueprints with "ring relations". More precisely, consider the functor -⊗ F 1 Z : Mon 0 → Ring which is the left adjoint to the forgetful functor, and consider the category ( -⊗ F 1 Z)/Ring. We shall denote by Z -Blp the full subcategory of ( -⊗ F 1 Z)/Ring formally defined in the same way as the subcategory blueprints Blp of -⊗ F 1 N/SRing. Analogously, one defines the category Z -Blp. A Z -B-scheme is then a scheme in Sch Z -Blp that admits a Zariski cover by affine schemes in (Z -Blp) op . We shall adopt hereafter the following terminological convention.

Convention 5.5
In what follows, by B-scheme we mean a Z -B-scheme, and by F 1scheme with relations we mean the combination of a Z -B-scheme and an F 1 -scheme in the sense of Definition 5.3. Now, Definition 4.16 and Definition 5.1 imply that, for every F 1 -scheme with relations ( , Z , Z , ), there is a natural transformation 1 : Z → Z given by the composition which will be called the first transferring map determined by the given F 1 -scheme with relations. As its name would suggest, the natural transformation 1 , loosely speaking, conveys information on about how many "points" of Z are compatible with the B-scheme that generates the pair ( , Z ). Actually, there is a different way to "transfer" this information from the B-scheme to the F 1 -scheme associated with the fibered object ( , Z , Z , ).
The counit of the adjunction -⊗ F 1 Z | -| induces a map Moreover, the natural transformation 4.20 induces a map Let B be the sheaf on the category Ring obtained as the pullback of the maps 5.2 and 5.3, i.e.
By composing the vertical arrow on the left with , we get a natural transformation which will be called the second transferring map determined by the F 1 -scheme ( , Z , Z , ).
In the case of an F 1 -scheme ( , Z , ), the natural transformation induces an isomorphism (|K|) Z (K) for every field K. Since for the finite field F q , one has |F q | = F 1 q−1 , it immediately follows, as observed in [1], that there is a bijective correspondence between the set of F q -points of Z and the set of F 1 q−1 -points of ; in others words, one has This result can be extended to our setting in two different ways, because, for a Bscheme underlying an F 1 -scheme with relations, we can think of its "F 1 q−1 -points" in two different senses. On the one hand, the forgetful functor | -| : Ring → Mon 0 admits the obvious factorization (cf. eq. 4.5). Clearly, one has and ρ(G Z (F q )) = |F q | = F 1 q−1 . Now, by definition, the first transferring map 1 factorises as 1 = • . Since gives isomorphisms (of sets) when evaluated on fields and is always locally injective, it is immediate to prove the following result.
Proposition 5.6 Let ( , Z , Z , ) be an F 1 -scheme with relations. The first transferring map 1 : Z → Z , when evaluated on a field, gives an injective map (of sets). In particular, the set of G Z (F q )-points of the underlying B-scheme naturally injects into the set of F q -points of the scheme Z (which is isomorphic to the set of F 1 q−1 -points of the monoidal scheme ).
On the other hand, one has the immersion σ : Proof Since we can work locally, we assume that the underlying scheme is given by a monoid M, a ring R, and a map M ⊗ F 1 Z → R satisfying the usual conditions. An F 1 q−1 -point is given by a commutative square such that the arrow on the top is induced by a map M → F 1 q−1 . The datum of a generic commutative square as above is equivalent to the datum of an The fact that the map on the top has the required property is equivalent to the fact that the image of the point above through the restriction map We are now interested in the case where the F 1 n -points of the underlying monoidal scheme are counted by a polynomial in n. Some preliminary definitions and results are in order.
A monoidal scheme is said to be noetherian if it admits a finite open cover by representable subfunctors {Spec(A i )}, with each A i a noetherian monoid. Recall that, as it is proved in [10, Theorem 5.10 and 7.8], a monoid is noetherian if and only if it is finitely generated. This immediately implies that, for any prime ideal p ⊂ M, the localized monoid M p is noetherian and the abelian group M × p of invertible elements in M p is finitely generated.

Remark 5.8
Notice that, given an F 1 -scheme ( , Z , ), the fact that the monoidal scheme is noetherian does not entail that the scheme Z is noetherian as well. Let us consider, for instance, the affine F 1 -scheme given by Z[X , ε i ]/(ε 2 i ) → Z[X ], with i ∈ N. The monoidal scheme is noetherian, while the ascending chain of ideals . . . ⊂ (ε 0 , . . . , ε i ) ⊂ (ε 0 , . . . , ε i+1 ) ⊂ . . . does not have a maximal element. Observe that, as for the points of the classical scheme, the presence of the ε i 's is immaterial; hence, one has the required isomorphism Z[X ](|K|) Z[X , ε i ]/(ε 2 i )(K) for any field K.
Let the geometrical realization of the monoidal scheme . Following Connes-Consani's definition [1, p. 25], we shall say that is torsion-free if, for any x ∈ , the abelian group O × ,x is torsion-free. Proof Since is noetherian, the abelian group O × ,x is finitely generated by the remark above. So, if is also torsion-free, then O × ,x is free of rank N (x), and, for any finite group G with #G = n, we have # Hom(O × ,x , G) = n N (x) . For the converse, suppose there is a point x such that O × ,x is not torsion-free. Being noetherian, O × ,x decomposes as a product Z n × i∈{1,...m} Z n i . For each prime number p 0 not dividing any of the n 1 , . . . , n m , say p 0 > LCM (n 1 , . . . , n m ), the number of elements of Hom(O × ,x , Z p 0 ) is then p n 0 . Since there are infinitely many such prime numbers, were # Hom(O × ,x , Z p ) a polynomial in p, it would be the polynomial p n . Take now a prime number p 1 dividing n 1 ; in that case, the number of elements of Hom(O × ,x , Z p 1 ) is greater than p n 1 . In conclusion, # Hom(O × ,x , Z p ) cannot be a polynomial in p.
By Lemma 5.9, for each noetherian and torsion-free monoidal scheme , one can define the polynomial The following result is proved in [1] (Theorem 4.10, (1) and (2)).

Theorem 5.10
Let ( , Z , ) be an F 1 -scheme such that the monoidal scheme is noetherian and torsion-free. Then (1) # (F 1 n ) = P( , n); (2) for each finite field F q the cardinality of the set of points of the scheme Z that are rational over F q is equal to P( , q − 1).
Note that the last statement immediately follows from eq. 5.6, which holds true without any additional assumption on the monoidal scheme.
Let ( , Z , Z , ) be an F 1 -scheme with relations such that the underlying Bscheme is noetherian and torsion-free. We define the polynomial Proposition 5.14 In the above hypotheses one has the inequality Q( , n) ≤ P( , n).
Proof It is clear that Hom B (O × ,x , F 1 n ) ⊂ Hom(O × ,x , F 1 n ), since the first set contains only the monoid morphisms that are compatible with the blueprint structure locally defined around x.
Remark 5. 15 Connes and Consani developed, in their papers [2] and [3], an approach based on -sets, that generalizes their previous theory of F 1 -schemes. Since the category of -sets is endowed with a natural monoidal closed structure, one can apply to that framework the general formalism introduced by Toën and Vaquié in [27]. In [3], a notion of scheme is defined; this notion is compared to that arising from [27], and the two are shown to be different. The situation is thus analogous to that occurring in the case of blue schemes, as described in [18]. Therefore, it seems worth briefly commenting upon Connes-Consani's construction.
Recall that a -set is a functor and all the other data correspond to associativity and commutativity conditions. On the other hand, by using the monoidal structure of the category of -spaces, one can define a second operation (say, multiplicative) on M.
Connes-Consani's basic idea is that it is possible to obtain more general structures on M by dropping the "special" condition; in particular, one can get a multiplicative monoid structure as above (but without the addition defined in Segal's setting). In order to model such a monoid structure, the only relevant information in the -set has to be the image M of 1 + ; technically, this is implemented by asking the image of an object n + , which is just the n fold coproduct of 1 + with itself, to be just the n fold coproduct of M with itself. It is shown in [3] that, by following a procedure of this kind, the categories Mon and Ring embed in the category -sets, as well as does the category MR as defined in [1]. In the case of rings (realized as objects in the category -sets), the corresponding schemes, according to Connes-Consani's definition, coincide with the classical ones [3,Prop. 7.9], whilst this is not the case for the schemes obtained by applying Toën-Vaquié's formalism [3,Lemma 8.1].
As a general consideration, we could say that the notion of scheme defined [27] places greater emphasis on the overall category, while that defined in [3] focuses more on the intrinsic geometric properties of each single object.