Finite presheaf topoi

hora v5 · 30 Oct 2026 Public

AI-assisted (Codex, Claude) · Verification (self-reported): Not confirmed

Answer proposed

Partial answer. For a presheaf $X$ let $X_{\mathrm{fin}}\subseteq X$ consist of the elements that generate a finite-valued subpresheaf; $X\mapsto X_{\mathrm{fin}}$ is the left exact coreflection onto locally finite presheaves (for a free monoid it is $h_*$ of Hora, Topoi of automata I, arXiv:2411.06358, Lemma 2.20). Every map from a finite presheaf into $X$ lands in $X_{\mathrm{fin}}$ , and a finite presheaf representing $\operatorname{Hom}(-,X)$ on $\mathbf{fPSh}(\mathcal C)$ is necessarily $X_{\mathrm{fin}}$ . Hence: (1) With $\Omega$ and $G^F$ computed in all presheaves, $\mathbf{fPSh}(\mathcal C)$ has a subobject classifier iff $\Omega_{\mathrm{fin}}$ is finite-valued, and has an exponential of $G$ by $F$ iff $(G^F)_{\mathrm{fin}}$ is finite-valued; these are then the universal objects. (2) It is a topos iff for every object $c$ and finite $X$ only finitely many $R\subseteq y(c)\times X$ have, for every $d$ , finitely many pullbacks $(f\times X)^*R$ along $f:d\to c$ . (3) If all hom-sets are finite, it is a topos iff every finite presheaf on each slice $\mathcal C/c$ has finitely many subpresheaves. Essentially finite slices are not necessary: for an infinite group $G$ and $M=G\sqcup\{0\}$ , $\mathbf{fPSh}(M)$ is a topos ( $M$ has three right ideals and $\operatorname{Sub}(M\times X)$ has at most $4^{|X|}$ elements), although $\operatorname{End}(0)=M$ in $M/{\ast}$ . A subobject classifier is not enough: on nonempty finite sets $\Omega$ is finite-valued, but the presheaf of equivalence relations has the subpresheaves “at most $k$ classes” for all $k$ , so it has no power object. For the monoid $(\mathbb N,+)$ there is no subobject classifier, since every ideal $n+\mathbb N$ lies in $\Omega_{\mathrm{fin}}$ (nor is it cartesian closed: Notes on Rieg Theory, Example 4.1.8). Open: a criterion in terms of $\mathcal C$ alone, local cartesian closedness, and whether for finite hom-sets the condition in (3) forces essentially finite slices. Not yet verified by a human.

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 Jérémie Marquès.

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.

Connections

Related: Rieg theory: the research programme
Built on by: A right Kan extension approach to finite presheaves

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