Motivation for the definition of an infinitesimal object
Solution 1:
In my view, what should be emphasised is that for an infinitesimal object $D$, $(-)^D : \mathcal{C} \to \mathcal{C}$ preserves all colimits. This makes the connection with other notions of smallness clearer.
Indeed, in algebraic categories such as $\mathbf{Set}$, $\mathbf{Ab}$, and $\mathbf{CRing}$, an object $A$ is finitely presentable if and only if the functor $\mathrm{Hom} (A, -) : \mathcal{C} \to \mathbf{Set}$ preserves filtered colimits. Moreover:
- In $\mathbf{Set}$, $\mathrm{Hom} (A, -) : \mathbf{Set} \to \mathbf{Set}$ preserves all colimits if and only if $A$ is a singleton. (In other words, the only infinitesimal object in $\mathbf{Set}$ is the point – as expected!)
- In $\mathbf{Ab}$, $\mathrm{Hom} (A, -) : \mathbf{Ab} \to \mathbf{Ab}$ preserves all colimits if and only if $A$ is a finitely generated projective $\mathbb{Z}$-module, which happens if and only if $A$ is a finitely generated free abelian group.
So there is a well-established precedent for thinking of $A$ as a very small object if (some version of) $\mathrm{Hom} (A, -)$ preserves all colimits.
That said, there are sometimes unexpected examples of infinitesimal objects. Let $\mathcal{B}$ be a small category with finite products. Then $\mathcal{C} = [\mathcal{B}^\mathrm{op}, \mathbf{Set}]$ is a cartesian closed category, with exponentials defined as follows: $$Y^X (B) = \mathrm{Hom} (\mathcal{B} (-, B) \times X, Y)$$ Hence, for $X = \mathcal{B} (-, A)$, $$Y^X (B) = \mathrm{Hom} (\mathcal{B} (-, B) \times \mathcal{B} (-, A), Y) \cong \mathrm{Hom} (\mathcal{B} (-, B \times A), Y) \cong Y (B \times A)$$ where in the last step we used the Yoneda lemma. Therefore every representable presheaf on $\mathcal{B}$ is infinitesimal. Conversely, now only assuming that $\mathcal{B}$ has a terminal object, if $X$ is infinitesimal, then $$Y^X (1) = \mathrm{Hom} (\mathcal{B} (-, 1) \times X, Y) \cong \mathrm{Hom} (X, Y)$$ so $\mathrm{Hom} (X, -) : [\mathcal{B}^\mathrm{op}, \mathbf{Set}] \to \mathbf{Set}$ preserves all colimits, so $X$ must be a retract of a representable presheaf on $\mathcal{B}$. Therefore, if $\mathcal{B}$ is a small idempotent-complete category with finite products, then a presheaf on $\mathcal{B}$ is infinitesimal if and only if it is representable.