This question is a follow-up of Extending functors defined on dense subcategories.

Let $\mathcal{K}$ be a locally presentable category. An object $X$ of $\mathcal{K}$ is called a Yoneda object if the functor $\mathcal{K}(X,-):\mathcal{K} \to \text{Set}$ is colimit-preserving.

For example, the representable functors $\mathcal{A}(a,-)$ of $\text{Set}^\mathcal{A}$, where $\mathcal{A}$ is a small category are Yoneda objects since $\text{Set}^\mathcal{A}(\mathcal{A}(a,-),F)=F(a)$. A Yoneda object is finitely presentable.

Question 1: Are there other Yoneda objects in $\text{Set}^\mathcal{A}$ than the representable functors (up to isomorphisms of functors) ?

Every locally presentable category is a reflective subcategory of a category of the form $\text{Set}^\mathcal{A}$.

Question 2: Let $\mathcal{K}$ be a locally presentable category having a strong generator of Yoneda objects. Is this category equivalent to a category of the form $\text{Set}^\mathcal{A}$ ?

  • 1
    $\begingroup$ Not an answer, but when I first met tiny objects I found enlightening two things. (i) in an abelian category regarded as unenriched there can't be tiny objects. (ii) the geometric meaning of a tiny object is precisely the fact that it is "so small it must fall into a component of any colimit": click $\endgroup$ – Fosco Loregian Jun 29 '17 at 20:31

The usual name for "Yoneda objects" is "tiny" or "small-projective".

  1. In general the tiny objects in a presheaf category are the retracts of representables. In particular, if $A$ is Cauchy-complete, then every tiny object is representable.

  2. Yes. In fact, instead of local presentability it is enough to assume cocompleteness; see Theorem 5.26 of Basic Concepts of Enriched Category Theory.

  • $\begingroup$ You beat me to it, although I have some literature references to add. $\endgroup$ – Todd Trimble Jun 29 '17 at 18:28
  • $\begingroup$ I mark your answer because you are the first one; and thanks to Jiri and Todd, it is very instructive. $\endgroup$ – Philippe Gaucher Jun 29 '17 at 18:59

These objects are usually called absolutely presentable. In $Set^\mathcal A$, they are precisely retracts of representables. Presheaf categories are characterized as cocomplete categories having a strong generator consisting of absolutely presentable objects (M. Bunge 1969).


Question 1: The Yoneda objects are precisely retracts of representables. There are quite a few buzzwords which are relevant here (Cauchy completion, idempotent splitting completion, Karoubi envelope, tiny objects, essential points), but they all describe the category of bicontinuous functors $Set^A \to Set$. You can find discussion in an article by Borceux and DeJean here; see particularly Proposition 2.

Question 2: If the tiny or Yoneda objects are dense in $\mathcal{K}$, meaning that the canonical map

$$\int^{a \in \text{Tiny}(\mathcal{K})} \mathcal{K}(a, k) \cdot a \to k$$

is an isomorphism for each object $k$ of $\mathcal{K}$, then $\mathcal{K}$ is equivalent to the presheaf topos $[\text{Tiny}(\mathcal{K})^{op}, Set]$. This result was first given in Marta Bunge's thesis as far as I know. As Mike said, being cocomplete and an atomic category is enough.


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.