# Is the unit compact in a closed monoidal (co)complete category?

Let $$\mathcal{M}$$ be a complete and cocomplete closed symmetric monoidal category, with unit $$I.$$

Does the functor $$\text{Hom}_{\mathcal{M}}(I,-)$$ always preserve filtered colimits?

A reference or a proof is the same for me. I believe this should be true as it's true in many examples such as $$\mathbf{Sets},$$ categories of modules, presheaf toposes etc.

• Do you mean the internal hom or the Set-valued hom? Jan 5, 2022 at 22:31
• @Zhen Lin hi. I mean the Set-valued Hom, as in the definition of compact/small object.
– user1011965
Jan 5, 2022 at 22:50

Let $$I$$ be a set. The category $$\textbf{Set}^I$$ is a presheaf topos, and in particular, complete and cocomplete and cartesian closed. The terminal object $$1$$ is not, in general, finitely presentable. Indeed, a general object $$X$$ in $$\textbf{Set}^I$$ is just an $$I$$-indexed family of sets, and the set of morphisms $$1 \to X$$ is in natural bijection with $$\prod_{i \in I} X_i$$.
Proposition. The functor $$X \mapsto \prod_{i \in I} X_i$$ preserves filtered colimits if and only if $$I$$ is a finite set.
For the "only if" direction, observe that $$X \cong \varinjlim_{J \in \mathscr{P}_\textrm{f} (I)} X_{(J)}$$, where $$\mathscr{P}_\textrm{f} (I)$$ is the set of finite subsets of $$I$$ and $$(X_{(J)})_i = X_i$$ if $$i \in J$$ and $$(X_{(J)})_i = \emptyset$$ if $$i \notin J$$. If $$I$$ is infinite, then $$\prod_{i \in I} (X_{(J)})_i$$ is always empty, hence $$\varinjlim_J \prod_{i \in I} (X_{(J)})_i$$ is also empty. But $$\prod_{i \in I} X_i$$ may be non-empty. So $$X \mapsto \prod_{i \in I} X_i$$ does not preserve filtered colimits if $$I$$ is infinite.　◼