Finite presheaf topoi

Question

For a small category $\mathcal C$, when is the category

$$\mathbf{fPSh}(\mathcal C)=[\mathcal C^{\mathrm{op}},\mathbf{FinSet}]$$

of pointwise finite-valued presheaves an elementary topos, or locally cartesian closed?

The motivation is enumerative combinatorics and Burnside riegs. This question was discussed with Jeremie Marques.

Known sufficient condition

Finite limits are computed pointwise. If every slice $\mathcal C/c$ is essentially finite, then the usual presheaf subobject classifier and exponentials are finite-valued, so $\mathbf{fPSh}(\mathcal C)$ is an elementary topos. This includes finite categories and groups regarded as one-object categories.

Here are the two relevant ambient formulas. The value $\Omega(c)$ is the set of sieves on $c$, so the ambient classifier is finite-valued exactly when there are finitely many sieves on each object. For finite-valued $F,G$,

$$(G^F)(c)=\operatorname{Nat}(y(c)\times F,G),$$

so the ambient exponential is finite-valued exactly when these sets are finite for every $c$.

These criteria describe when the ambient constructions stay in the finite-valued subcategory. They are not, without an additional argument, a necessary-and-sufficient classification of when that subcategory has its own classifier and exponentials.

Remaining question and next step

Find a general criterion beyond essentially finite slices. For a proposed criterion, distinguish closure under the ambient presheaf constructions from existence of the required universal objects inside the finite-valued category.

Note 8 records the right Kan extension approach. Its first missing ingredient is the precise adjunction or comonad that would implement the reduction.

Source

Original problem 6.0.1.

@misc{sumire:2609.00007:5:v4,
  author = {hora},
  title = {{Finite presheaf topoi}},
  year = {2026},
  note = {SUMIRE:2609.00007/5, v4},
  url = {https://sumire.aiformath.jp/p/2609.00007/n/5/v/4}
}