Comparison of polygraphic homologies for arbitrary strict k-categories

Determine whether, for every strict k-category C, the canonical natural morphism from the polygraphic homology of the underlying strict infinity-category to the k-polygraphic homology of C is an isomorphism.

Background

For a strict k-category C, the paper compares two derived homology theories. One is obtained by regarding C as a strict infinity-category and applying ordinary polygraphic homology; the other is obtained by computing polygraphic homology directly in the category of strict k-categories. A canonical natural morphism connects these two constructions.

The paper proves that this comparison morphism is an isomorphism when C is a 1-category regarded as a strict k-category for k≥1, and when C is a groupoid regarded as a strict k-category for k≥0. The unresolved problem is the corresponding comparison for general strict k-categories outside these cases. This formulation records the question as stated before the two classes of positive results are given.