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A locally small category $\mathscr C$ is called petite if, for every functor $F : \mathscr C \to \mathscr D$ with locally small codomain, and for every object $D \in \mathscr D$, the presheaf $\mathscr D(F{-}, D) : \mathscr C^{\text{op}} \to \mathrm{Set}$ is small [DL23].

If $\mathscr C$ is essentially small, then every presheaf on $\mathscr C$ is small, so that every essentially small category is petite.

Is the converse true? That is, is a petite category necessarily essentially small?

For context, an affirmative answer would give a purely 2-categorical way to characterise the essentially small categories within the 2-category of locally small categories.

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  • $\begingroup$ Haven’t really thought much about this but it would surely be related to the definition of “small” from the original “Yoneda Structures” paper — if memory serves, the proof of correctness of this definition is in another standalone paper. $\endgroup$
    – Tim Campion
    Commented Jul 2 at 18:05
  • $\begingroup$ Ok note to self the relevant characterization is that $C$ is small iff $C$ and $Fun(C^{op},Set)$ are locally small, which apparently wasn’t proven in the literature until more than a decade after it was implied in Street and Walters. On the size of Categories Freyd and Street, TAC in 1991. And somehow I didn’t find this cited in the paper you link to. Anyway, have you thought about adapting Freyd and Street’s work to your setting? $\endgroup$
    – Tim Campion
    Commented Jul 2 at 18:30
  • $\begingroup$ @TimCampion: it is not clear that Freyd and Street's characterisation holds for any example other than ordinary categories (see this question, for instance). On the other hand, if Di Liberti–Loregian's definition captures smallness, I would expect it to immediately generalise to enriched and internal categories. $\endgroup$
    – varkor
    Commented Jul 2 at 18:43
  • $\begingroup$ (There is another distinction, which is that Freyd and Street's characterisation is relevant to the 2-category of large categories, whereas Di Liberti–Loregian's is relevant to the 2-category of locally small categories.) $\endgroup$
    – varkor
    Commented Jul 2 at 18:47
  • $\begingroup$ @TimCampion: I don’t follow your last comment; $\mathcal{D}(F-,D)$ is certainly set-valued for all $F$ as in the question regardless of $\mathcal{C}$, since $\mathcal{D}$ is assumed locally small. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Jul 5 at 10:43

1 Answer 1

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It seems to me that the following are equivalent for a locally small category $B$:

  1. $B$ is petite
  2. every presheaf $q\colon B^{\mathrm{op}} \to \mathsf{Set}$ is small
  3. for every presheaf $q$ and copresheaf $p\colon B \to \mathsf{Set}$ the coend $\int^{y \in B} py \times qy$ is small
  4. the presheaf category $\mathsf{Set}^{B^{\mathrm{op}}}$ is locally small
  5. $B$ is essentially small

$\phantom{}$ 4. $\Rightarrow$ 5. is Freyd and Street's result.

$\phantom{}$ 3. $\Rightarrow$ 4. is explained in Example 5.8 here

$\phantom{}$ 2. $\Rightarrow$ 3.: $q$ being small means the existence of a small full subcategory $B' \subseteq B$ with $q \cong \int^{y' \in B'} B(-, y') \times qy'$. Hence we have bijections of (potentially large) sets

$$ \int^{y \in B} py \times qy \cong \int^{y \in B} py \times \int^{y' \in B'} B(y, y') \times qy' \\\cong \int^{y' \in B'} \left(\int^{y \in B} py \times B(y, y') \right) \times qy' \cong \int^{y' \in B'} py' \times qy'$$

Since the last coend here involves only small sets it is small. Hence so is the first coend, as required.

$\phantom{}$ 1. $\Rightarrow$ 2.: Regard $q$ as a profunctor $B^{\textrm{op}} \times \{*\} \to \mathsf{Set}$ and take its cograph $[q]$. It has objects $\textrm{ob}([q]) = \textrm{ob}(B) \sqcup \{*\}$, is locally small, and has an embedding $i\colon B \hookrightarrow [q]$ such that $[q](i-, *) \cong q$. If $B$ is petite then the left-hand side is small so that $q$ is too.

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    $\begingroup$ Very nice argument! $\endgroup$
    – varkor
    Commented Jul 6 at 9:25

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