Left-Exact Characterization of Enriched Topoi

Definitions and mathematical statement

Fix an enrichment base $V$ in the setting intended by the source, and let $E$ be a presentable $V$-category. The supplied record does not specify the hypotheses on $V$, the meaning of enriched presentability, or the definition of a $V$-topos; these must be recovered from the source before the conjecture has a self-contained formulation.

Write $\mathcal{P}$ for the free cocompletion construction, as in the source, and write \[ \operatorname{colim}_E\colon \mathcal{P}(E)\longrightarrow E \] for its stated colimit functor. Left exactness means preservation of finite limits in the intended setting. The record does not specify the relevant notion of finite limit or the size conventions under which this construction and functor are defined.

Conjecture (André Joyal, joint work with Mathieu Anel). For every presentable $V$-category $E$ satisfying the source's intended ambient hypotheses, \[ E\text{ is a }V\text{-topos} \quad\Longleftrightarrow\quad \operatorname{colim}_E\colon\mathcal{P}(E)\longrightarrow E \text{ is left exact}. \] This is a source-relative statement: the unspecified hypotheses and definitions above remain unresolved, rather than being additional established assumptions supplied by this draft.

The source also connects this conjecture to the existence of a (pseudo) distributive law \[ Q_{\omega}\mathcal{P}\longrightarrow\mathcal{P}Q_{\omega}, \] where $\mathcal{P}\colon\mathrm{VCAT}\to\mathrm{VCAT}$ is the free cocompletion monad and $Q_{\omega}\colon\mathrm{VCAT}\to\mathrm{VCAT}$ is the free finite completion monad. The ambient categorical structure denoted by $\mathrm{VCAT}$, the precise distributive-law axioms, and the nature of this connection are not specified in the supplied record. In particular, the record does not assert an equivalence between existence of such a law and the displayed conjecture.

Short English statement

The conjecture asks whether a presentable category enriched over $V$ is a $V$-topos exactly when its colimit functor from its free cocompletion preserves finite limits. The enrichment hypotheses and the precise meanings of $V$-topos, presentability, and finite limits require clarification from the cited source. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30006035$, source problem code catalogue entry, title: "Left-Exact Characterization of Enriched Topoi."
• Supplied source citation: Homotopical Algebra and Higher Structures (2025), Oberwolfach Reports, pp. 2251--2308, DOI 10.4171/owr/2024/39. Supplied source URL: https://doi.org/10.4171/owr/2024/39. The record attributes the contribution "A research program for higher V-topoi" to André Joyal, joint work with Mathieu Anel; its proposed year is 2024.
• Supplied literature source: "Oberwolfach Report 39 (2024)," https://doi.org/10.4171/owr/2024/39. The report documents workshop discussions and open problems. The supplied citation uses 2025, while the report label, DOI, and proposed year refer to 2024; this bibliographic distinction has not been resolved here.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open, states that the characterization appears as a conjecture in the report, and reports that no subsequent proof of the full assertion was located. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.