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 $\simeq \mathbb{R}^n$.
In the broader context of synthetic differential geometry there may exist spaces — in a smooth topos $\mathcal{T}$ with line object $R$ — considerably more general than manifolds. While for all of them there is a notion of tangent bundle $T X : = X^D$ (sometimes called a synthetic tangent bundle, with $D$ the infinitesimal interval), not all such tangent bundles necessarily have $R$-linear fibers!
A microlinear space is essentially an object $X$ in a smooth topos, such that its tangent bundle does have $R$-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.
(microlinear space)
Let $\mathcal{T}$ be a smooth topos with line object $R$. An object $X \in \mathcal{T}$ is a microlinear space if for each diagram $\Delta : J \to \mathcal{T}$ of infinitesimal spaces in $\mathcal{T}$ and for each cocone $\Delta \to \Delta_c$ under it such that homming into $R$ produces a limit diagram , $R^\Delta_c \simeq \lim_{j \in J} R^{\Delta_j}$, also homming into $X$ produces a limit diagram: $X^\Delta_c \simeq \lim_{j \in J} X^{\Delta_j}$.
The main point of this definition is the following property.
(fiberwise linearity of tangent bundle)
For every microlinear space $X$, the tangent bundle $X^D \to X$ has a natural fiberwise $R$-module-structure.
We describe first the addition of tangent vectors, then the $R$-action on them and then prove that this is a module-structure.
Addition With $D = \{\epsilon \in R| \epsilon^2 = 0 \}$ the infinitesimal interval and $D(2) = \{(\epsilon_1, \epsilon_2) \in R \times R | \epsilon_i^2 = 0\}$ we have a cocone
such that
is a limit cone, by the Kock-Lawvere axiom satisfied in the smooth topos $\mathcal{T}$. Since $X$ is microlinear, also the canonical map
is an isomorphism. With $\Id \times Id : D \to D(2)$ the diagonal map, we then define the fiberwise addition $X^D \times_X X^D \to X^D$ in the tangent bundle $X^D$ to be given by the map
On elements, this sends two elements $v_1, v_2 \in X^D$ in the same fiber to the element $v_1 + v_2$ of $X^D$ given by the map $(v_1 + v_2) : d \mapsto r^{-1}(v_1,v_2)(d,d)$.
Multiplication $\cdot : R \times X^D \to X^D$ is defined componentwise by
$(\alpha \cdot v) : d \mapsto v (\alpha \cdot d)$.
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 $(\mathcal{T},R)$ we have the following.
The standard line $R$ is microlinear.
The collection of microlinear spaces is closed under limits in $\mathcal{T}$:
for $X = \lim_i X_i$ a limit of microlinear spaces $X_i$, also $X$ is microlinear.
Mapping spaces into microlinear spaces are microlinear: for $X$ any microlinear space and $\Sigma$ any space, also the internal hom $X^\Sigma$ is microlinear.
This is obvious from the standard properties of limits and the fact that the internal hom-functor $(-)^Y : \mathcal{T} \to \mathcal{T}$ preserves limits. (See limits and colimits by example if you don’t find it obvious.)
by definition
Let $\Delta$ be the tip of a cocone $\Delta_j$ of infinitesimal spaces such that $\lim_j R^{\Delta_j} = R^\Delta$. Then
with $\Delta$ as above we have (writing $[A,B]$ for the internal hom otherwise equivalently denoted $B^A$)
(microlinear loci)
Let $\mathcal{F}$, $\mathcal{G}, \mathcal{Z}, \mathcal{B}$ be the smooth toposes of the same name that are discussed in detail in MSIA, capter III. These are constructed there as categories of sheaves on a subcategory of the category $\mathbb{L} = (C^\infty Ring^{fin})$ of smooth loci.
All representable objects in these smooth toposes are microlinear.
For $\mathcal{F}$ and $\mathcal{G}$ this is the statement of MSIA, chapter V, section 7.1.
For $\mathcal{Z}$ and $\mathcal{B}$ the argument is similarly easy:
These are categories of sheaves on the full category $\mathbb{L} = (C^\infty Ring^{fin})^{op}$. The line object $R$ is representable in each case, $R = \ell C^\infty(\mathbb{R})$. Every object in $\mathbb{L}$ is a limit (not necessarily finite) over copies of $R$ in $\mathbb{L}$. Accordingly, every object $\ell A$ of $\mathbb{L}$ satisfies the microlinearity axioms in $\mathbb{L}$ in that for each cocone $\Delta \to \Delta_c : J \to \mathbb{L}$ of infinitesimal objects such that $R^{\Delta_c} \simeq \lim_{j \in J} R^{\Delta_j}$ we also have $(\ell A)^{\Delta_c} \simeq \lim_{j \in J} (\ell A)^{\Delta_j}$. Now, the Yoneda embedding $Y : \mathbb{L} \to PSh(C)$ preserves limits and exponentials. Since the Grothendieck topology in question is subcanonical, $Y((\ell A)^{\Delta_j})$ is in $Sh(C)$ and hence is the exponential object $Y(\ell A)^{Y \Delta_j}$ there. Finally, the finite limit over $J$ is preserved by the reflection $PSh(C) \to Sh(C)$ (sheafification, which acts trivially on our representables), so $Y(\ell A)^{\Delta_c} \simeq \lim_{j \in J} Y(\ell A)^{Y(\Delta_j)}$ and hence all $Y(\ell A)$ are microlinear in $\mathcal{Z}$ and $\mathcal{B}$.
The notion of microlinear space in the above fashion is due to
and was studied further under the name strong infinitesimal linearity
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