In his books Synthetic Differential Geometry and Synthetic Geometry of Manifolds, Anders Kock defines a microlinear space, or an infinitesimally linear space $M$ as one with the following property.
Microlinearity. For each $n\geq 1$ and each $n$-tuple of maps $t_i:D\rightarrow M$ satisfying $t_i(0)=t_j(0)$ there's a unique $\ell:D(n)\rightarrow M$ satisfying $\ell \circ \iota_i =t_i$.
Here $D= \left\{x\in R\mid x^2=0 \right\}$, $D(n)= \left\{ (x_1,\dots,x_n)\in R^n\mid x_i\cdot x_j=0\;\forall i,j \right\}$, and $\iota_i:d\mapsto (0,\dots, d,\dots ,0)$.
What is the geometric meaning of this condition? Can anyone shed light on it? I understand it's a condition on all $n$-tuples of tangent vectors with common basepoint, I just don't get what it's saying. This MSE question is also related.
It's more-or-less saying that any $n$ tangent vectors span an infinitesimal linear space. In other words, it's a smoothness condition. For example, given $x, y : D \to M$ such that $x (0) = y (0)$, we get $z : D (2) \to M$ such that $z (d, 0) = x (d)$ and $z (0, d) = y (d)$; but for any $(\lambda, \mu) \in R^2$, we also have $\lambda x + \mu y : D \to M$ given by $d \mapsto z (\lambda d, \mu d)$. The uniqueness condition ensures that this is all well defined and gives the tangent space an $R$-module structure.
For comparison, consider a non-smooth space such as the union of the two coordinate axes in the plane. In that case, you can pick two linearly independent (... in some sense ...) tangent vectors at the origin, but we can't "linearly interpolate" between them.