Synthetic differential geometry
In traditional differential geometry a smooth manifold may be thought of as a “locally linear space”: a space that is locally isomorphic to a vector space .
In the broader context of synthetic differential geometry there may exist spaces — in a smooth topos with line object — considerably more general than manifolds. While for all of them there is a notion of tangent bundle (sometimes called a synthetic tangent bundle, with the infinitesimal interval), not all such tangent bundles necessarily have -linear fibers!
A microlinear space is essentially an object in a smooth topos, such that its tangent bundle does have -linear fibers.
In fact the definition is a bit stronger than that, but the main point in practice of microlinearity is that the linearity of the fibers of the tangent bundle allows to apply most of the familiar constructions in differential geometry to these spaces.
Let be a smooth topos with line object . An object is a microlinear space if for each diagram of infinitesimal spaces in and for each cocone under it such that homming into produces a limit diagram , , also homming into produces a limit diagram: .
The main point of this definition is the following property.
(fiberwise linearity of tangent bundle)
For every microlinear space , the tangent bundle has a natural fiberwise -module-structure.
Construction and Proof
We describe first the addition of tangent vectors, then the -action on them and then prove that this is a module-structure.
Addition With the infinitesimal interval and we have a cocone
is a limit cone by the Kock-Lawvere axiom. Since is microlinear, also the canonical map
is an isomorphism. With the diagonal map, we then define the fiberwise addition in the tangent bundle to be given by the map
On elements, this sends two elements in the same fiber to the element of given by the map .
Multiplication is defined componentwise by
One checks that this is indeed unital, associative and distributive.
A large class of examples is implied by the following proposition.
(closedness of the collection of microlinear spaces)
In every smooth topos we have the following.
The standard line is microlinear.
The collection of microlinear spaces is closed under limits in :
for a limit of microlinear spaces , also is microlinear.
Mapping spaces into microlinear spaces are microlinear: for any microlinear space and any space, also the internal hom is microlinear.
This is obvious from the standard properties of limits and the fact that the internal hom-functor preserves limits. (See limits and colimits by example if you don’t find it obvious.)
Let be the tip of a cocone of infinitesimal spaces such that . Then
with as above we have (writing for the internal hom otherwise equivalently denoted )
For and this is the statement of MSIA, chapter V, section 7.1.
For and the argument is similarly easy:
These are categories of sheaves on the full category . The line object is representable in each case, . Every object in is a limit (not necessarily finite) over copies of in . Accordingly, every object of satisfies the microlinearity axioms in in that for each cocone of infinitesimal objects such that we also have . Now, the Yoneda embedding preserves limits and exponentials. Since the Grothendieck topology in question is subcanonical, is in and hence is the exponential object there. Finally, the finite limit over is preserved by the reflection (sheafification, which acts trivially on our representables), so and hence all are microlinear in and .
The notion of microlinear space in the above fashion is due to
and was studied further under the name strong infinitesimal linearity
- Anders Kock, R. Lavendhomme, Strong infinitesimal linearity, with applications to string difference and affine connections, Cahiers de Top. 25 (1984)
This is similar to but stronger than the earlier “condition (E)” given in
which apparently was also called “infinitesimal linearity” (without the “strong”).
Spaces satisfying this condition were called infinitesimally linear spaces, for instance in
The later re-typing of that book
contains in its appendix D the definition of microlinearity as above.
A comprehensive discussion of microlinearity is in chapter V, section 1 of