Transfer Operations in Homology of Finite Categories

Definitions and mathematical statement

Let $\mathcal C$ denote a finite category. Conventionally, this means that its sets of objects and morphisms are finite; the supplied source does not explicitly specify its finiteness convention. Write $H_*(\mathcal C;M)$ and $H^*(\mathcal C;M)$ as placeholders for the intended homology and cohomology with coefficient data $M$. The source does not specify the homology or cohomology theory, the coefficient data, or their variance, so this notation does not select a particular theory.

Source question. For the homology and cohomology of finite categories, how can restriction, corestriction or transfer, conjugation, and inflation be formulated, and what properties should a categorical transfer map satisfy?

The requested task is to formulate these operations, including their domains, codomains, hypotheses, and compatibility properties. However, the source specifies neither the functors or other categorical data inducing the operations nor the required properties of transfer. It therefore supplies a research question rather than a definite universally quantified conjecture. No particular construction, existence assertion, or list of transfer axioms can be extracted without additional choices.

Short English statement

The problem asks for a theory of restriction, transfer, conjugation, and inflation for the homology and cohomology of finite categories. The underlying theories, coefficients, admissible categorical maps, and required transfer identities are unspecified, so a precise conjecture remains to be formulated.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004487$, title Transfer Operations in Homology of Finite Categories, source problem code catalogue entry. The original, clean, and displayed statements agree. No individual proposer is named; the supplied proposed year is 2020.
• Supplied report attribution: Cohomology of Finite Groups: Interactions and Applications (2021), Oberwolfach Reports, pp. 1197--1261, DOI 10.4171/owr/2020/23. Supplied source link: https://doi.org/10.4171/owr/2020/23.
• Supplied literature source: OWR Cohomology of Finite Groups report, https://ems.press/content/serial-article-files/46862.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and describes the fragment as requesting a general theory without a precise conjecture, hence unresolved and unclear. This is supplied dated curation, not independently verified. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement exact and describes it as clear, self-contained, publication-ready, and retained byte-for-byte. That supplied verification description does not resolve the missing mathematical specifications identified in the supplied literature assessment.