nLab Z-functor

This page is about the concept in functorial algebraic geometry. For functors between Z-categories see there.

Context

Algebra

Constructivism, Realizability, Computability

Contents

Definition

A β„€\mathbb{Z}-functor or integers functor is a functor from the category of commutative rings CRing to the category of sets Set. The category of β„€\mathbb{Z}-functors is thus the functor category Set CRing\mathrm{Set}^\mathrm{CRing}.

Examples

  • The forgetful functor 𝔸 1:CRingβ†’Set\mathbb{A}^1:\mathrm{CRing} \to \mathrm{Set} is a β„€\mathbb{Z}-functor.

  • More generally, any affine scheme is a β„€\mathbb{Z}-functor.

Properties

Definition

Let XX be a β„€\mathbb{Z}-functor. Then the ring of fractions π’ͺ(X)\mathcal{O}(X) of XX is the set X⇒𝔸 1X \Rightarrow \mathbb{A}^1 of natural transformations from XX to the forgetful functor 𝔸 1\mathbb{A}^1.

The ring structure on X⇒𝔸 1X \Rightarrow \mathbb{A}^1 is defined pointwise by the following, for every commutative ring RR and element x∈X(R)x \in X(R):

0 R(x)≔01 R(x)≔10_R(x) \coloneqq 0 \qquad 1_R(x) \coloneqq 1
(a R+b R)(x)≔a R(x)+b R(x)(a_R + b_R)(x) \coloneqq a_R(x) + b_R(x)
(βˆ’a R)(x)β‰”βˆ’a R(x)(- a_R)(x) \coloneqq - a_R(x)
(a Rβ‹…b R)(x)≔a R(x)β‹…b R(x)(a_R \cdot b_R)(x) \coloneqq a_R(x) \cdot b_R(x)

References

The terminology β€œβ„€\mathbb{Z}-functor” is due to

popularized through the English translation of the first couple of chapters:

Formulation in univalent homotopy type theory:

Last revised on October 29, 2024 at 10:25:58. See the history of this page for a list of all contributions to it.