nLab reduced object

Idea

In a context of synthetic differential geometry/differential cohesion a reduced object is one “without purely infinitesimal extension”.

For instance in the context of formal schemes/formal smooth manifolds the reduced objects are the genuine schemes and the genuine smooth manifold, those without formal extension.

Accordingly, an anti-reduced object is one consisting entirely of infinitesimal extension, hence is an infinitesimally thickened point.

Beware that reduced objects in general do “contain infinitesimals in between their classical points”, in that not every map from an anti-reduced object into them is necessarily constant. The objects “without any infinitesimals” in the sense that all such maps are constant are instead the coreduced objects.

Definition

A context of differential cohesion is determined by the existence of an adjoint triple of modalities

$$\Re \dashv \Im \dashv \& \,,$$

where $\Re$ and $\&$ are idempotent comonads and $\Im$ is an idempotent monad.

A reduced object or reduced type is one in the full subcategory defined by the leftmost modality $\Re$.

Examples

In smooth differential geometry

Consider the Cahiers topos

$$\mathbf{H} = Sh(\{\mathbb{R}^n \times Spec(W)\}_{n,W})$$

as a model of synthetic differential geometry/differential cohesion (where $W$ denotes Weil algebras/Artin algebras). The sub-topos of reduced obects

$$\mathbf{H}_{reduced} \hookrightarrow \mathbf{H}$$

is the topos of smooth spaces

$$\mathbf{H}_{reduced} = Sh(\{\mathbb{R}^n\}) \,.$$

An object $D \in \mathbf{H}$ with is an anti-reduced object, hence whose reduction coreflection is the terminal object, $\Re(D) \simeq \ast$ is an infinitesimally thickened point.

For instance the formal dual $D^1 = Spec(\mathbb{R}[\epsilon](\epsilon^2))$ of the ring of dual numbers is such that its reduction is the point $\Re(D) \simeq \ast$.

Under Yoneda embedding every smooth manifold is in $\mathbf{H}_{reduced}$ and is hence a reduced object in $\mathbf{H}$. More generally there are formal smooth manifolds in $\mathbf{H}$ and they are generally not reduced.

For example for $\Sigma \in SmoothMfd \hookrightarrow \mathbf{H}_{reduced}\hookrightarrow \mathbf{H}$ an ordinary smooth manifold (hence reduced) the object

$$\Sigma \times D^1 \in \mathbf{H}$$

is a formal smooth manifold which is not reduced. It reduction is

$$\Re(\Sigma\times D^1) \simeq \Sigma \,.$$

In particular the real line which is the smooth line object of the smooth topos $\mathbf{H}$

$$\mathbb{R}^1 \in \mathbf{H}_{reduced}\hookrightarrow \mathbf{H}$$

is reduced, $\Re(\mathbb{R}) \simeq \mathbb{R}$.

Observe from these examples that reduced objects do “contain infinitesimal points in between their classical points”, which just means that there are non-constant morphisms of the form

$$D \longrightarrow \Sigma \,.$$

Contrast between reduced and coreduced objects

The objects $X \in \mathbf{H}$ for which all maps out of anti-reduced objects $D$ are constant maps are instead the coreduced objects.

The coreduced objects are the ones with “no infinitesimal behavior”, and the reduced objects are the ones “whose infinitesimal behavior is determined by their non-infinitesimal behavior”. A reduced object does contain infinitesimal points; what it lacks are “purely infinitesimal directions” while a coreduced object has no infinitesimal points.

In algebraic geometry

A reduced scheme is one all whose local rings of functions have no non-zero nilpotent elements.

Related concepts

• infinitesimal space, infinitesimally thickened point

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

  $(\esh \dashv \flat \dashv \sharp )$

  • discrete object, codiscrete object, concrete object

  • points-to-pieces transform

  • structures in cohesion

• dR-shape modality$\dashv$ dR-flat modality

  $\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

• classical modality

tangent cohesion

• differential cohomology diagram

differential cohesion

• (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

  $(\Re \dashv \Im \dashv \&)$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

graded differential cohesion

• fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality

  $(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

singular cohesion

$$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$$

• formally smooth morphism, formally etale morphism, formally unramified morphism