Contents

# Contents

## Idea

An anti-reduced object or simple infinitesimal type is one whose reduction is the point, hence one consisting entirely of “infinitesimal extension”, i.e. an infinitesimally thickened point.

## Definition

In the context of differential cohesion, an anti-reduced obect is an comodal type $X$ for the infinitesimal shape modality $\Im$

$\Im(X) \simeq \ast \,.$

## Examples

### Formal moduli problems

In homotopy type theory/higher topos theory anti-reduced types are essentially what is also called “formal moduli problems” (these are typically required to satisfy one more condition besides being anti-reduced, namely being infinitesimally cohesive in the sense of Lurie).

cohesion

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

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

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

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

tangent cohesion

differential cohesion

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