Operational Objecthood as Separable Sharing

This work develops an operational account of objecthood. Instead of treating an object merely as an element of a category, it asks when a candidate can be formed and preserved as one re-identifiable object under multiple descriptions, operations, and categorical transitions. The central claim is that objecthood has two layers. First, an object is formed through separable sharing. Its descriptive axes must remain separately operable while jointly referring to the same candidate. Categorically, if A1,…,AnA_1,,A_nA1,…,An are attribute or operand categories and each projects to a common reference category BBB, then the object-level category is not the unrestricted product A1×⋯×AnA_1×⋯× A_nA1×⋯×An, but the pullback-like compatibility locus Aobj=A1×BA2×B⋯×BAn.Aobj = A_1×_B A_2×_B⋯×_B A_n.Aobj=A1×BA2×B⋯×BAn. Thus an object is not a mere tuple of independent attributes. It is a compatible family of readings that share one reference. Second, once formed, the object must be preserved under admissible operations and categorical transitions. An operational categorical system is written as Ci=(Ai,Oi), C_i=( A_i, O_i),Ci=(Ai,Oi), where Ai A_iAi is a category of operands and Oi⊆End⁡(Ai) O_i⊆ End( A_i)Oi⊆End(Ai) is a chosen category of admissible endofunctors. For two such systems, an operand transition Tij:Ai→AjTᵢⱼ: A_i→ A_jTij:Ai→Aj and an operator transition Φij:Oi→OjΦᵢⱼ: O_i→ O_jΦij:Oi→Oj preserve objecthood when, for every admissible operator O∈OiO∈ O_iO∈Oi, there is a natural isomorphism ηO:Tij∘O⇒∼Φij(O)∘Tij.η_O: Tᵢⱼ∘ O {~} Φᵢⱼ(O)∘ Tᵢⱼ.ηO:Tij∘O∼Φij(O)∘Tij. This condition says that acting first and transferring later agrees, up to natural isomorphism, with transferring first and acting by the transferred operator. The paper interprets familiar categorical structures—pullbacks, naturality, functoriality, and action preservation—as criteria for operational objecthood. Examples include ordinary objects such as coins, physical mixtures such as saltwater, Euler’s formula as a compatibility relation preserving unit rotation, and object detection in artificial intelligence, where pixels, labels, bounding boxes, and latent features count as one object only when they jointly stabilize the same candidate. The final thesis is: objecthood=separable sharing+stable re-identification.{ objecthood = separable sharing + stable re-identification. }objecthood=separable sharing+stable re-identification. In ordinary language, a thing becomes an object when its many distinguishable ways of being read still point back to one same candidate, and its admissible ways of being acted upon continue to preserve that candidate.