nLab reduced object




Discrete and concrete objects

Modalities, Closure and Reflection



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.


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.


In smooth differential geometry

Consider the Cahiers topos

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

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

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

is the topos of smooth spaces

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

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

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

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

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

Σ×D 1H \Sigma \times D^1 \in \mathbf{H}

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

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

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

1H reducedH \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

DΣ. D \longrightarrow \Sigma \,.

Contrast between reduced and coreduced objects

The objects XHX \in \mathbf{H} for which all maps out of anti-reduced objects DD 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.


infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 }

Last revised on October 9, 2017 at 14:14:39. See the history of this page for a list of all contributions to it.