Let's call a category with arbitrary coproducts a $\coprod$-category. A $\coprod$-functor is a functor which preserves coproducts. An example is $\mathsf{Set}$, and this is in fact the universal $\coprod$-category equipped with an object, namely $\{1\}$. By this I mean that for every $\coprod$-category $\mathcal{C}$ there is an equivalence of categories $$\mathrm{Hom}_{\coprod}(\mathsf{Set},\mathcal{C}) \simeq \mathcal{C},~ F \mapsto F(\{1\}).$$ Question. What is the universal $\coprod$-category equipped with an object $A$ and a morphism $A \to A$? More generally, if $S$ is any set, what is the universal $\coprod$-category equipped with an object and an $S$-indexed family of morphisms $(A \to A)_{s \in S}$?

I am pretty sure that sets are still the objects, resp. the copowers $X \otimes A_{\mathrm{univ}} = \bigoplus_{x \in X} A_{\mathrm{univ}}$. But the morphisms are more complicated.


The category with small coproducts freely generated by a category $\mathcal{C}$ is $\mathbf{Fam} (\mathcal{C})$, the category of families of objects in $\mathcal{C}$. The quickest description is as follows: $\mathbf{Fam} (\mathcal{C})$ is the category obtained by applying the Grothendieck construction to the (contravariant) functor $X \mapsto \mathcal{C}^X$. In other words:

  • The objects in $\mathbf{Fam} (\mathcal{C})$ are pairs $(X, A)$ where $X$ is a set and $A$ is an object in $\mathcal{C}^X$, i.e., an $X$-indexed family of objects in $\mathcal{C}$.
  • The morphisms $(X, A) \to (Y, B)$ in $\mathbf{Fam} (\mathcal{C})$ are pairs $(f, h)$ where $f$ is a map $X \to Y$ and $h$ is a morphism $A \to f^* B$ in $\mathcal{C}^X$, i.e., for each $x \in X$, a morphism $h_x : A_x \to B_{f (x)}$.
  • Composition and identities are the obvious ones.

To verify that $\mathbf{Fam} (\mathcal{C})$ has the required universal property, one uses the fact that the canonical comparison map $$\prod_{x \in X} \coprod_{y \in Y} \mathcal{C} (A_x, B_y) \to \mathbf{Fam} (\mathcal{C}) ((X, A), (Y, B))$$ is a bijection. (This is essentially the same argument that shows that $\mathbf{Ind} (\mathcal{C})$ is the category with filtered colimits freely generated by $\mathcal{C}$.)

Incidentally, when $\mathcal{C}$ is locally small, there is a small-coproduct-preserving functor $\mathbf{Fam} (\mathcal{C}) \to [\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ that extends the Yoneda embedding. The natural bijection above implies this is a fully faithful embedding. The presheaves in the essential image are said to be familially representable. Thus $\mathbf{Fam} (\mathcal{C})$ is equivalent to the full subcategory of $[\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ spanned by those presheaves that are familially representable, i.e., the presheaves that are small coproducts of representables. (Again, this is entirely analogous to $\mathbf{Ind} (\mathcal{C})$.)

Thus, the category with small coproducts freely generated by an endomorphism is equivalent to a certain full subcategory of the category of sets equipped with an $\mathbb{N}$-action.

  • $\begingroup$ Great answer! Thank you! :) Of course, an object $(X,A)$ should be seen as $\coprod_{x \in X} A$. $\endgroup$ – Martin Brandenburg Jun 28 '15 at 13:01
  • $\begingroup$ Regarding the last paragraph: These are precisely the free $\mathbb{N}$-sets. $\endgroup$ – Martin Brandenburg Jun 28 '15 at 17:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.