$I$-Chain Models of Spaces in Coalgebras and Lie Algebras

Definitions and mathematical statement

For terminology, relative to a chosen commutative ground ring $k$, a differential graded cocommutative coalgebra consists of a chain complex $C$ of $k$-modules with a coassociative chain map $\Delta:C\to C\otimes_k C$ satisfying $\tau\circ\Delta=\Delta$, where $\tau(x\otimes y)=(-1)^{|x||y|}y\otimes x$ for homogeneous elements. A differential graded Lie algebra over $k$ is a chain complex with a degree-zero graded Lie bracket for which the differential is a graded derivation. The source does not specify the ground ring, grading convention, or additional coalgebra conditions such as counitality or conilpotence; these definitions do not select those missing hypotheses.

Source question. Can $I$-chains be used to construct models of spaces in the settings of differential graded cocommutative coalgebras and differential graded Lie algebras?

Unresolved formulation. The supplied record does not define $I$-chains or their indexing category, specify the class of spaces, or define the required meaning of a model. In particular, it does not state the comparison functors, equivalences, or recovery properties that such models must satisfy, nor whether both algebraic settings must be related by a single construction. Thus the question is preserved as a research target; the supplied information does not determine a fully quantified existence statement.

The supplied dated assessment says that homotopy colimits of $I$-chains produce $E_\infty$-coalgebras and that a known obstruction to the expected left-induced model structures prevents a complete Lie or cocommutative-coalgebra model of the requested kind. The record does not identify the precise obstruction or its hypotheses. This assessment is supplied curation, not an independently verified theorem or a claim that every possible construction is impossible.

Short English statement

The question asks whether $I$-chains can provide algebraic models of spaces using differential graded cocommutative coalgebras and Lie algebras. The intended spaces, coefficients, construction, and notion of equivalence remain unspecified. The supplied open-status label and obstruction assessment require independent review.

Sources

• Local UnsolvedMath registry: permanent ID $30004847$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied record labels the problem open and its statement corrected_verified; these are registry annotations, not independent mathematical verification.
• Supplied source citation: Homotopical Algebra and Higher Structures (2022), Oberwolfach Reports, pp. 2497--2560, DOI 10.4171/owr/2021/46. Literature-source label: Oberwolfach Report 46. https://doi.org/10.4171/owr/2021/46. The record gives proposed year 2021.
• The original statement appends the text "Canonical $E_\infty$-operads involved in homotopy colimits of $I$-chain complexes" and the name Birgit Richter. This preserves the supplied attribution context; the separate proposer field is empty.
• Supplied literature source: Comonoidal properties of I-chains. https://www.math.uni-hamburg.de/en/personen/richter/ichains100822.pdf.
• Literature assessment checked at 2026-08-22: the supplied curation labels the question open and reports the $E_\infty$-coalgebra construction and obstruction described above. This is supplied dated literature triage derived from the OWR curation, not independently verified source consultation or a current literature assessment.