synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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.
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. For if $(-)^\Delta$ preserves colimits, its right adjoint is
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
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).
William Lawvere, Toward the Description in a Smooth Topos of the Dynamically Possible Motions and Deformations of a Continuous Body, Cah.Top.Géom.Diff.Cat. 21 no.4 (1980) pp.377-392. (pdf)
William Lawvere, Left and right adjoint operations on spaces and data types , Theor. Comp. Sci. 316 (2004) pp.105-111.
Ieke Moerdijk, Gonzalo Reyes, Models for Smooth Infinitesimal Analysis , Springer Heidelberg 1991. (appendix 4)
Anders Kock, Gonzalo E. Reyes, Aspects of Fractional Exponent Functors , TAC 5 (1999) pp.251-265. (pdf)
David Yetter, On Right Adjoints of Exponential Functors , JPAA 45 (1987) pp.287-304. (Corrections in JPAA 58 (1989) pp.103-105)
William Lawvere, Categorical algebra for continuum micro physics, Journal of Pure and Applied Algebra 175 (2002) 267–287