Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

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

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Compact objects

objects $d \in C$ such that $C(d,-)$ commutes with certain colimits

# Contents

## Definition

William Lawvere‘s definition of an atomic infinitesimal space is as an object $\Delta$ in a topos $\mathcal{T}$ such that the inner hom functor $(-)^\Delta : \mathcal{T} \to \mathcal{T}$ has a right adjoint (is an atomic object).

Notice that by definition of inner hom, $(-)^\Delta$ always has a left adjoint. A right adjoint can only exist for very particular objects. Therefore the term amazing right adjoint.

## Right adjoints to representable exponentials

Assume $\mathcal{T} = Sh(C)$ is a Grothendieck topos, that the Grothendieck topology on the site $C$ is subcanonical. Let $\Delta \in C \hookrightarrow Sh(C)$ be a representable object. Then, $(-)^\Delta: Sh(C)\to Sh(C)$ has a right adjoint, hence $\Delta$ is an atomic infinitesimal space, precisely if $(-)^\Delta: C\to C$ preserves colimits. This is a special case of the general adjoint functor theorem#in_toposes. For if $(-)^\Delta$ preserves colimits, its right adjoint is

$(-)_\Delta : (Y \in Sh(C)) \mapsto (U \mapsto Sh_C(U^\Delta, Y)) \,.$

The $Y_\Delta$ defined this way is indeed a sheaf, due to the assumption that $(-)^\Delta$ preserves colimits. So this is indeed a right adjoint.

• A topos $\mathcal{X}$ is a local topos (over Set) if its global section functor $\Gamma = Hom(1_{\mathcal{X}}, -)$ admits a right adjoint. This is hence an “external” version of the amazing right adjoint, exhibiting $1_{\mathcal{X}}$ as “atomic”.

• The topic of amazing right adjoints appears to have been studied mostly for toposes, but a related definition has been given for any category (with products): if $O$ is an exponentiable object of a category $\mathcal{C}$, then $O$ being a tiny object in the sense of Definition 0.1 means that the endofunctor $(-)^O$ has a right adjoint. (This adjoint is sometimes denoted $(-)^{1/O}$, c.f. Lawvere, p.269)

In this situation, one then has, symbolically

$(-)\times O \quad \dashv\quad (-)^O \quad \dashv\quad (-)^{1/O}$

## References

The ubiquity of right adjoints to exponential functors in the context of synthetic differential geometry was first pointed out in Lawvere (1980). Lawvere (2004) suggests to augment lambda calculus with such fractional operators. Thorough discussion of the concept is in Yetter (1987) and Kock&Reyes (1999). Moerdijk&Reyes (1991) have a succinct overview in the context of SDG as does Lawvere (1997).

Last revised on July 21, 2017 at 02:47:11. See the history of this page for a list of all contributions to it.