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)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(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)
In a context of differential cohesion, with infinitesimal shape modality $\Im$, then for every object $X \in \mathbf{H}$ its infinitesimal disk bundle $T_{inf}X$ is the bundle over $X$ whose fiber over a point is the infinitesimal neighbourhood of that point, fomalized as the homotopy fiber at this point of the unit of the infinitesimal shape modality $\Im$ on $X$.
The collection of these fiber hence forms the homotopy fiber product
of the $X$ component $i \colon X \to \Im X$ of the unit of the infinitesimal shape modality $\Im$ with itself.
Conversely, by the pasting law, the fibers of $p \colon T_{inf}X \to X$ over global points of $X$ are indeed the infinitesimal disks around these points.
Evidently $T_{inf}X$ is the first stage in the Cech nerve of $X \to \Im(X)$, hence the object of morphisms of the groupoid object corresponding to this effective epimorphism. By the discussion at Lie algebroid – General abstract definition this is an infinity-Lie algebroid, namely the (possibly higher jet order) tangent Lie algebroid of $X$.
More generally, for $(E \to X) \in \mathbf{H}_{/X}$ a bundle over $X$, then $T_{inf}E \coloneqq T_{inf} X \times_{\Im X} E \simeq X\times_{\Im X} E$, sitting in the pasting composite of pullbacks
Stated more abstractly, this means that forming infinitesimal disk bundles is the monad
induced by the adjoint triple of base change along $i$
In the standard models of differential cohesion (such as for formal smooth infinity-groupoids), $\Im X$ is the standard de Rham stack of $X$ obtained by identifying infinitesimal neighbours, and so then $T_{\inf }X$ is the formal neighbourhood of the diagonal of $X$, in the traditional sense. Indeed, in these standard models $X \to \Im X$ is a 1-epimorphism, hence effective, and so on 0-truncated $X$ the above pullback equivalently equibits the de Rham stack $\Im X$ for 0-truncated $X$ as the coequalizer of the two projections out of the formal neighbourhood of the diagoal, which is the traditional definition of $\Im X$.
The tangent complex of a derived algebraic stack $X$ is equivalently the (sheaf of modules of) sections of the formal neighbourhood of the diagonal of $X$ (Hennion 13). Hence by the above one may generally think of (sections of) $T_{inf}X$ as being the tangent complex of $X$.
The infinitesimal disk bundle construction is the left adjoint monad to the jet comonad
The underlying adjunction has been observed in Kock 1980, prop. 2.2, in the context of synthetic differential geometry .
In terms of differential cohesion this pair of adjoint monads is seen in Khavkine & Schreiber 2017, p. 23 as the adjoint pair induced by the base change adjoint triple:
For $X$ a $V$-manifold, then its infinitesimal disk bundle is a fiber bundle (fiber infinity-bundle) with typical fiber $\simeq \mathbb{D}^V_e$. This is the associated bundle (associated infinity-bundle) to the frame bundle $Fr(X) \to X$ (or more generally of the higher order frame bundle when $(\Re \dashv \Im)$ encodes higher order infinitesimal thickening).
See at differential cohesion – frame bundles.
Discussion in synthetic differential geometry is, under the name “bundles of $k$-monads”, in
Anders Kock, above prop. 2.2 of: Formal manifolds and synthetic theory of jet bundles, Cahiers de Topologie et Géométrie Différentielle Catégoriques 21 3 (1980) [numdam:CTGDC_1980__21_3_227_0]
Anders Kock, p. 39 of Synthetic Geometry of Manifolds, 2009 (pdf)
Discussion in differential cohesion is in
and formalization in homotopy type theory in
Relation to the tangent complex is discussed in
Last revised on August 12, 2023 at 16:45:04. See the history of this page for a list of all contributions to it.