Math preprint finds a nerve that changes under thinification

A mathematical preprint reports a sharp split in a formal construction: the Roberts–Street nerve of the walking coinductive equivalence is not contractible in the homotopy theory of saturated complicial sets, yet every n-th thinification of that nerve is contractible when $n \ge 0$. The abstract also states that reflecting the example to a left higher category produces a contractible object.

The document is labeled an arXiv version v1 dated 28 Aug 2026.

What is being tested

The analyzed object is a marked simplicial set obtained as the Roberts–Street nerve of the walking coinductive equivalence. Its underlying omega-category, called E, has two objects and is contractible. The nerve and E are related constructions, but the paper’s central finding is that they do not share the same contractibility status in the homotopy theory under study.

Here, “contractible” is a technical homotopy-theoretic term. The paper expresses the contractible outcome by saying that a thinified nerve maps to a point object through a weak equivalence, the formal comparison used for this purpose. For a general reader, that means the theory treats the construction as having the same relevant homotopy-level shape as a point.

The question is whether the nerve of E is contractible in its own right, and what happens to that answer after truncation or thinification. The paper also examines how reflection changes the description.

The operation that changes the answer

The first result is a classification: the Roberts–Street nerve of E is a saturated complicial set. The paper makes its contractibility comparison inside the homotopy theory of such objects.

Thinification is the operation that produces the indexed versions of the nerve examined in the paper. Its result is exact and uniform: for every index n satisfying $n \ge 0$, the n-th thinification of the Roberts–Street nerve is contractible in the saturated-complicial homotopy theory.

The paper states a broader theorem. For every contractible omega-category P and every index n satisfying $n \ge 0$, the corresponding thinified Roberts–Street nerve maps to the point object by a weak equivalence. In plain language, after this operation the formal comparison used by the theory identifies the result with the point case.

That result should not be confused with a claim that the original nerve was contractible. The paper establishes the opposite for the unmodified nerve. The central contrast is therefore between the nerve before thinification, which is non-contractible, and every indexed version after thinification, which is contractible.

A change of viewpoint

Reflection supplies a second way to state the distinction. Under the model identification used in the paper, the reflector is computed levelwise by thinification. The abstract states that reflection to a left higher category is contractible, while the original Roberts–Street nerve remains non-contractible in the saturated-complicial homotopy theory.

The proof is formal and uses the suspension–hom adjunction as a Quillen adjunction for saturated complicial sets. This is the technical framework the paper uses for its homotopy-theoretic comparison.

The right-versus-left interpretation carries an explicit condition: it depends on the stated model identification. The result is consequently specific to the formal object examined here, its nerve, and the thinification or reflection operations described in the paper.

Paper data and sources

Original title: About the contractibility of the walking coinductive equivalence
Authors: Viktoriya Ozornova, Martina Rovelli, Tashi Walde
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text