Motivation for the definition of an infinitesimal object

Question

An infinitesimal object $D$ in a Cartesian closed category $\mathsf{C}$ is one for which the internal Hom functor $$(-)^D: \mathsf{C} \to \mathsf{C}$$ has a right adjoint.

I am wondering what is the motivation for this definition? Also, how would one interpret this right adjoint and what it does, especially in the context of topology and (generalized) differential geometry?

With regards to my background, I know the basics of category theory from Awodey's text, and differential geometry at the level of John Lee's series. I have an interest in synthetic geometry but I am just beginning to read Kock's text on the topic.

Answer 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.