Last time we have seen how an affine $\mathbb{C} $-algebra R gives us a **maxi-functor** (because the associated sets are typically huge)

$\mathbf{maxi}(R)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto Hom_{\mathbb{C}-alg}(R, \mathbb{C} A) $

Substantially smaller sets are produced from finitely generated $\mathbb{Z} $-algebras S (therefore called **mini-functors**)

$\mathbf{mini}(S)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto Hom_{\mathbb{Z}-alg}(S, \mathbb{Z} A) $

Both these functors are ‘represented’ by existing geometrical objects, for a maxi-functor by the complex affine variety $X_R = \mathbf{max}(R) $ (the set of maximal ideals of the algebra R) with complex coordinate ring R and for a mini-functor by the integral affine scheme $X_S = \mathbf{spec}(S) $ (the set of all prime ideals of the algebra S).

The ‘philosophy’ of F_un mathematics is that an object over this virtual field with one element $\mathbb{F}_1 $ records the essence of possibly complicated complex- or integral- objects in a small combinatorial thing.

For example, an n-dimensional complex vectorspace $\mathbb{C}^{n} $ has as its integral form a lattice of rank n $\mathbb{Z}^{\oplus n} $. The corresponding $\mathbb{F}_1 $-objects only records the dimension n, so it is a finite set consisting of n elements (think of them as the set of base-vectors of the vectorspace).

Similarly, all base-changes of the complex vectorspace $\mathbb{C}^n $ are given by invertible matrices with complex coefficients $GL_n(\mathbb{C}) $. Of these base-changes, the only ones leaving the integral lattice $\mathbb{Z}^{\oplus n} $ intact are the matrices having all their entries integers and their determinant equal to $\pm 1 $, that is the group $GL_n(\mathbb{Z}) $. Of these integral matrices, the only ones that shuffle the base-vectors around are the permutation matrices, that is the group $S_n $ of all possible ways to permute the n base-vectors. In fact, this example also illustrates Tits’ original motivation to introduce $\mathbb{F}_1 $ : the finite group $S_n $ is the Weyl-group of the complex Lie group $GL_n(\mathbb{C}) $.

So, we expect a geometric $\mathbb{F}_1 $-object to determine a much smaller functor from finite abelian groups to sets, and, therefore we call it a nano-functor

$\mathbf{nano}(N)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto N(A) $

but as we do not know yet what the correct geometric object might be we will only assume for the moment that it is a subfunctor of some mini-functor $\mathbf{mini}(S) $. That is, for every finite abelian group A we have an inclusion of sets $N(A) \subset Hom_{\mathbb{Z}-alg}(S,\mathbb{Z} A) $ in such a way that these inclusions are compatible with morphisms. Again, take pen and paper and you are bound to discover the correct definition of what is called a **natural transformation**, that is, a ‘map’ between the two functors $\mathbf{nano}(N) \rightarrow \mathbf{mini}(S) $.

Right, now to make sense of our virtual F_un geometrical object $\mathbf{nano}(N) $ we have to connect it to properly existing complex- and/or integral-geometrical objects.

Let us define a **gadget** to be a couple $~(\mathbf{nano}(N),\mathbf{maxi}(R)) $ consisting of a nano- and a maxi-functor together with a ‘map’ (that is, a natural transformation) between them

$e~:~\mathbf{nano}(N) \rightarrow \mathbf{maxi}(R) $

The idea of this map is that it visualizes the elements of the set $N(A) $ as $\mathbb{C} A $-points of the complex variety $X_R $ (that is, as a collection of $o(A) $ points of $X_R $, where $o(A) $ is the number of elements of $A $).

In the example we used last time (the forgetful functor) with $N(A)=A $ any group-element $a \in A $ is mapped to the algebra map $\mathbb{C}[x,x^{-1}] \rightarrow \mathbb{C} A~,~x \mapsto e_a $ in $\mathbf{maxi}(\mathbb{C}[x,x^{-1}]) $. On the geometry side, the points of the variety associated to $\mathbb{C} A $ are all algebra maps $\mathbb{C} A \rightarrow \mathbb{C} $, that is, the $o(A) $ **characters** ${ \chi_1,\ldots,\chi_{o(A)} } $. Therefore, a group-element $a \in A $ is mapped to the $\mathbb{C} A $-point of the complex variety $\mathbb{C}^* = X_{\mathbb{C}[x,x^{-1}]} $ consisting of all character-values at $a $ : ${ \chi_1(a),\ldots,\chi_{o(A)}(g) } $.

In mathematics we do not merely consider objects (such as the gadgets defined just now), but also the morphisms between these objects. So, what might be a morphism between two gadgets

$~(\mathbf{nano}(N),\mathbf{maxi}(R)) \rightarrow (\mathbf{nano}(N’),\mathbf{maxi}(R’)) $

Well, naturally it should be a ‘map’ (that is, a natural transformation) between the nano-functors $\phi~:~\mathbf{nano}(N) \rightarrow \mathbf{nano}(N’) $ together with a morphism between the complex varieties $X_R \rightarrow X_{R’} $ (or equivalently, an algebra morphism $\psi~:~R’ \rightarrow R $) such that the extra gadget-structure (the evaluation maps) are preserved.

That is, for every finite Abelian group $A $ we should have a commuting diagram of maps

$\xymatrix{N(A) \ar[rr]^{\phi(A)} \ar[d]^{e_N(A)} & & N'(A) \ar[d]^{e_{N’}(A)} \\ Hom_{\mathbb{C}-alg}(R,\mathbb{C} A) \ar[rr]^{- \circ \psi} & & Hom_{\mathbb{C}-alg}(R’,\mathbb{C} A)} $

Not every gadget is a F_un variety though, for those should also have an integral form, that is, define a mini-functor. In fact, as we will see next time, an affine $\mathbb{F}_1 $-variety is a gadget determining a unique mini-functor $\mathbf{mini}(S) $.

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