A category for bijective combinatorics

Referee: [Definition of the cobordism category (likely §2 or §3)] The central claim requires that composition of cobordisms of signed sets is well-defined, associative, and functorial while preserving the correspondence to matchings. The manuscript must supply an explicit definition of how signs are tracked under composition (including any cancellation rules) and prove that the resulting structure is a category; without this, the extension asserted in the abstract cannot be verified.

Authors: Section 2 defines the category Cob of cobordisms of signed sets. Objects are pairs (S,σ) with σ:S→{+,−}. A morphism from (A,α) to (B,β) is a finite collection of intervals whose endpoints carry the prescribed signs and whose interior points are unlabeled. Composition is realized by gluing along the common boundary signed set; at each glued point the signs must be opposite (+ with −) for the gluing to be admissible, which is the only cancellation rule. Identities are the “straight” cobordisms consisting of |S| intervals of length zero. Associativity follows because path concatenation is associative up to reparametrization, and the sign condition is local. The forgetful functor to the category of matchings between unsigned sets is defined by taking the net pairing after all admissible cancellations; this is shown to be well-defined on equivalence classes of cobordisms. We will insert a short subsection (new §2.3) that writes the composition operation and the verification of the category axioms in fully expanded form. revision: partial

Referee: [Chain-to-matching correspondence (likely the main theorem)] The correspondence theorem—that every chain of cobordisms between unsigned sets A and B canonically yields a matching A↔B—must be stated and proved without extra sign conditions or quotient relations. If the proof relies on implicit cancellations or additional equivalences, the claim that the construction “packages” the Garsia–Milne principle without reference to involutions is undermined.

Authors: Theorem 3.1 states precisely that any finite chain of cobordisms whose initial and terminal objects are unsigned sets A and B determines a unique matching A↔B. The proof is by induction on chain length: the base case is the identity cobordism, which yields the empty matching; the inductive step composes one additional cobordism and applies the sign-cancellation rule already present in the definition of composition. No auxiliary quotient or extra sign condition is imposed; the only identifications are those forced by the geometric gluing. Because the cancellations are performed inside the category operation rather than by constructing an auxiliary involution on a larger set, the construction indeed separates the Garsia–Milne principle from any explicit involution. We will add a one-paragraph expansion of the inductive step to make the absence of hidden relations completely transparent. revision: partial