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\begin{document}

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\section*{reduced object}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohesion}{}\paragraph*{{Cohesion}}\label{cohesion}

[[!include cohesive infinity-toposes - contents]]

\hypertarget{discrete_and_concrete_objects}{}\paragraph*{{Discrete and concrete objects}}\label{discrete_and_concrete_objects}

[[!include discrete and concrete objects - contents]]

\hypertarget{modalities_closure_and_reflection}{}\paragraph*{{Modalities, Closure and Reflection}}\label{modalities_closure_and_reflection}

[[!include modalities - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{in_smooth_differential_geometry}{In smooth differential geometry}\dotfill \pageref*{in_smooth_differential_geometry} \linebreak
\noindent\hyperlink{contrast_between_reduced_and_coreduced_objects}{Contrast between reduced and coreduced objects}\dotfill \pageref*{contrast_between_reduced_and_coreduced_objects} \linebreak
\noindent\hyperlink{in_algebraic_geometry}{In algebraic geometry}\dotfill \pageref*{in_algebraic_geometry} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

In a context of [[synthetic differential geometry]]/[[differential cohesion]] a \emph{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 map|constant]]. The objects ``without any infinitesimals'' in the sense that all such maps are constant are instead the [[coreduced objects]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

\begin{displaymath}
\Re \dashv \Im \dashv \&
  \,,
\end{displaymath}
where $\Re$ and $\&$ are [[idempotent monad|idempotent]] [[comonads]] and $\Im$ is an [[idempotent monad]].

A \textbf{reduced object} or \textbf{reduced type} is one in the [[full subcategory]] defined by the leftmost modality $\Re$.

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{in_smooth_differential_geometry}{}\subsubsection*{{In smooth differential geometry}}\label{in_smooth_differential_geometry}

Consider the [[Cahiers topos]]

\begin{displaymath}
\mathbf{H} = Sh(\{\mathbb{R}^n \times Spec(W)\}_{n,W})
\end{displaymath}
as a [[categorical semantics|model]] of [[synthetic differential geometry]]/[[differential cohesion]] (where $W$ denotes [[infinitesimally thickened point|Weil algebras]]/[[Artin algebras]]). The sub-topos of reduced obects

\begin{displaymath}
\mathbf{H}_{reduced} \hookrightarrow \mathbf{H}
\end{displaymath}
is the topos of [[smooth spaces]]

\begin{displaymath}
\mathbf{H}_{reduced} = Sh(\{\mathbb{R}^n\})
  \,.
\end{displaymath}
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 \emph{[[infinitesimally thickened point]]}.

For instance the [[Isbell duality|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

\begin{displaymath}
\Sigma \times D^1 \in \mathbf{H}
\end{displaymath}
is a [[formal smooth manifold]] which is not reduced. It reduction is

\begin{displaymath}
\Re(\Sigma\times D^1) \simeq \Sigma
  \,.
\end{displaymath}
In particular the [[real line]] which is the smooth line object of the [[smooth topos]] $\mathbf{H}$

\begin{displaymath}
\mathbb{R}^1 \in \mathbf{H}_{reduced}\hookrightarrow \mathbf{H}
\end{displaymath}
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 map|constant]] morphisms of the form

\begin{displaymath}
D \longrightarrow \Sigma
  \,.
\end{displaymath}
\hypertarget{contrast_between_reduced_and_coreduced_objects}{}\subsubsection*{{Contrast between reduced and coreduced objects}}\label{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.

\hypertarget{in_algebraic_geometry}{}\subsubsection*{{In algebraic geometry}}\label{in_algebraic_geometry}

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

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[infinitesimal space]], [[infinitesimally thickened point]]

\end{itemize}
[[!include cohesion - table]]

\begin{itemize}%
\item [[formally smooth morphism]], [[formally etale morphism]], [[formally unramified morphism]]

\end{itemize}
[[!redirects reduced objects]]

[[!redirects reduced type]] [[!redirects reduced types]]

[[!redirects reduced space]] [[!redirects reduced spaces]]



\end{document}