Cover Image

Operational Objecthood as Separable Sharing

Fuente: Zenodo

Saved in:

Bibliographic Details
Main Author:	Yeon, Jeong Min
Format:	Recurso digital
Language:	English
Published:	Zenodo 2026
Subjects:	Philosophy Mathematical logic
Online Access:	Acceder al recurso
Tags:	Add Tag No Tags, Be the first to tag this record!

_version_	1866901956614684672
author	Yeon, Jeong Min
author_facet	Yeon, Jeong Min
contents	<p>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.</p> <p>The central claim is that objecthood has two layers. First, an object is formed through <strong>separable sharing</strong>. Its descriptive axes must remain separately operable while jointly referring to the same candidate. Categorically, if <span class="katex"><span class="katex-mathml">A1,…,AnA_1,\dots,A_n</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span><span class="mpunct">,</span><span class="minner">…</span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span> are attribute or operand categories and each projects to a common reference category <span class="katex"><span class="katex-mathml">BB</span><span class="katex-html"><span class="base"><span class="mord mathnormal">B</span></span></span></span>, then the object-level category is not the unrestricted product <span class="katex"><span class="katex-mathml">A1×⋯×AnA_1\times\cdots\times A_n</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span><span class="mbin">×</span></span><span class="base"><span class="minner">⋯</span><span class="mbin">×</span></span><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span>, but the pullback-like compatibility locus</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">Aobj=A1×BA2×B⋯×BAn.A_{\mathrm{obj}} = A_1\times_B A_2\times_B\cdots\times_B A_n.</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">obj</span></span></span></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span><span class="mbin">×<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">B</span></span></span></span></span></span></span></span><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span><span class="mbin">×<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">B</span></span></span></span></span></span></span></span><span class="base"><span class="minner">⋯</span><span class="mbin">×<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">B</span></span></span></span></span></span></span></span><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p> <p>Thus an object is not a mere tuple of independent attributes. It is a compatible family of readings that share one reference.</p> <p>Second, once formed, the object must be preserved under admissible operations and categorical transitions. An operational categorical system is written as</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">Ci=(Ai,Oi),\mathcal C_i=(\mathcal A_i,\mathcal O_i),</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathcal">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mopen">(</span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathcal">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span></span></p> <p>where <span class="katex"><span class="katex-mathml">Ai\mathcal A_i</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span> is a category of operands and</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">Oi&sube;End⁡(Ai)\mathcal O_i\subseteq \operatorname{End}(\mathcal A_i)</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathcal">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mrel">&sube;</span></span><span class="base"><span class="mop"><span class="mord mathrm">End</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p> <p>is a chosen category of admissible endofunctors. For two such systems, an operand transition</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">Tij:Ai→AjT_{ij}:\mathcal A_i\to\mathcal A_j</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ij</span></span></span></span></span></span></span></span><span class="mrel">:</span></span><span class="base"><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mrel">→</span></span><span class="base"><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">j</span></span></span></span></span></span></span></span></span></span></span></p> <p>and an operator transition</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">Φij:Oi→Oj\Phi_{ij}:\mathcal O_i\to\mathcal O_j</span><span class="katex-html"><span class="base"><span class="mord">Φ<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ij</span></span></span></span></span></span></span></span><span class="mrel">:</span></span><span class="base"><span class="mord"><span class="mord mathcal">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mrel">→</span></span><span class="base"><span class="mord"><span class="mord mathcal">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">j</span></span></span></span></span></span></span></span></span></span></span></p> <p>preserve objecthood when, for every admissible operator <span class="katex"><span class="katex-mathml">O∈OiO\in\mathcal O_i</span><span class="katex-html"><span class="base"><span class="mord mathnormal">O</span><span class="mrel">∈</span></span><span class="base"><span class="mord"><span class="mord mathcal">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span>, there is a natural isomorphism</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">ηO:Tij∘O⇒&sim;Φij(O)∘Tij.\eta_O: T_{ij}\circ O \xRightarrow{\sim} \Phi_{ij}(O)\circ T_{ij}.</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">η</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">O</span></span></span></span></span></span></span><span class="mrel">:</span></span><span class="base"><span class="mord"><span class="mord mathnormal">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ij</span></span></span></span></span></span></span></span><span class="mbin">∘</span></span><span class="base"><span class="mord mathnormal">O</span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mrel mtight">&sim;</span></span></span></span></span></span></span></span><span class="base"><span class="mord">Φ<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ij</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">O</span><span class="mclose">)</span><span class="mbin">∘</span></span><span class="base"><span class="mord"><span class="mord mathnormal">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ij</span></span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p> <p>This condition says that acting first and transferring later agrees, up to natural isomorphism, with transferring first and acting by the transferred operator.</p> <p>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.</p> <p>The final thesis is:</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">objecthood=separable sharing+stable re-identification.\boxed{ \text{objecthood} = \text{separable sharing} + \text{stable re-identification}. }</span><span class="katex-html"><span class="base"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="boxpad"><span class="mord text">objecthood</span><span class="mrel">=</span><span class="mord text">separable sharing</span><span class="mbin">+</span><span class="mord text">stable re-identification</span>.</span></span></span></span></span></span></span></span></span></p> <p>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.</p>
format	Recurso digital
id	zenodo_https___doi_org_10_5281_zenodo_20069437
institution	Zenodo
language	eng
publishDate	2026
publisher	Zenodo
record_format	zenodo
spellingShingle	Operational Objecthood as Separable Sharing Yeon, Jeong Min Philosophy Mathematical logic <p>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.</p> <p>The central claim is that objecthood has two layers. First, an object is formed through <strong>separable sharing</strong>. Its descriptive axes must remain separately operable while jointly referring to the same candidate. Categorically, if <span class="katex"><span class="katex-mathml">A1,…,AnA_1,\dots,A_n</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span><span class="mpunct">,</span><span class="minner">…</span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span> are attribute or operand categories and each projects to a common reference category <span class="katex"><span class="katex-mathml">BB</span><span class="katex-html"><span class="base"><span class="mord mathnormal">B</span></span></span></span>, then the object-level category is not the unrestricted product <span class="katex"><span class="katex-mathml">A1×⋯×AnA_1\times\cdots\times A_n</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span><span class="mbin">×</span></span><span class="base"><span class="minner">⋯</span><span class="mbin">×</span></span><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span>, but the pullback-like compatibility locus</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">Aobj=A1×BA2×B⋯×BAn.A_{\mathrm{obj}} = A_1\times_B A_2\times_B\cdots\times_B A_n.</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">obj</span></span></span></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span><span class="mbin">×<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">B</span></span></span></span></span></span></span></span><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span><span class="mbin">×<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">B</span></span></span></span></span></span></span></span><span class="base"><span class="minner">⋯</span><span class="mbin">×<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">B</span></span></span></span></span></span></span></span><span class="base"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p> <p>Thus an object is not a mere tuple of independent attributes. It is a compatible family of readings that share one reference.</p> <p>Second, once formed, the object must be preserved under admissible operations and categorical transitions. An operational categorical system is written as</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">Ci=(Ai,Oi),\mathcal C_i=(\mathcal A_i,\mathcal O_i),</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathcal">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mopen">(</span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathcal">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span></span></p> <p>where <span class="katex"><span class="katex-mathml">Ai\mathcal A_i</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span> is a category of operands and</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">Oi&sube;End⁡(Ai)\mathcal O_i\subseteq \operatorname{End}(\mathcal A_i)</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathcal">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mrel">&sube;</span></span><span class="base"><span class="mop"><span class="mord mathrm">End</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p> <p>is a chosen category of admissible endofunctors. For two such systems, an operand transition</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">Tij:Ai→AjT_{ij}:\mathcal A_i\to\mathcal A_j</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ij</span></span></span></span></span></span></span></span><span class="mrel">:</span></span><span class="base"><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mrel">→</span></span><span class="base"><span class="mord"><span class="mord mathcal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">j</span></span></span></span></span></span></span></span></span></span></span></p> <p>and an operator transition</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">Φij:Oi→Oj\Phi_{ij}:\mathcal O_i\to\mathcal O_j</span><span class="katex-html"><span class="base"><span class="mord">Φ<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ij</span></span></span></span></span></span></span></span><span class="mrel">:</span></span><span class="base"><span class="mord"><span class="mord mathcal">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span><span class="mrel">→</span></span><span class="base"><span class="mord"><span class="mord mathcal">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">j</span></span></span></span></span></span></span></span></span></span></span></p> <p>preserve objecthood when, for every admissible operator <span class="katex"><span class="katex-mathml">O∈OiO\in\mathcal O_i</span><span class="katex-html"><span class="base"><span class="mord mathnormal">O</span><span class="mrel">∈</span></span><span class="base"><span class="mord"><span class="mord mathcal">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span>, there is a natural isomorphism</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">ηO:Tij∘O⇒&sim;Φij(O)∘Tij.\eta_O: T_{ij}\circ O \xRightarrow{\sim} \Phi_{ij}(O)\circ T_{ij}.</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">η</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">O</span></span></span></span></span></span></span><span class="mrel">:</span></span><span class="base"><span class="mord"><span class="mord mathnormal">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ij</span></span></span></span></span></span></span></span><span class="mbin">∘</span></span><span class="base"><span class="mord mathnormal">O</span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mrel mtight">&sim;</span></span></span></span></span></span></span></span><span class="base"><span class="mord">Φ<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ij</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">O</span><span class="mclose">)</span><span class="mbin">∘</span></span><span class="base"><span class="mord"><span class="mord mathnormal">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ij</span></span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p> <p>This condition says that acting first and transferring later agrees, up to natural isomorphism, with transferring first and acting by the transferred operator.</p> <p>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.</p> <p>The final thesis is:</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">objecthood=separable sharing+stable re-identification.\boxed{ \text{objecthood} = \text{separable sharing} + \text{stable re-identification}. }</span><span class="katex-html"><span class="base"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="boxpad"><span class="mord text">objecthood</span><span class="mrel">=</span><span class="mord text">separable sharing</span><span class="mbin">+</span><span class="mord text">stable re-identification</span>.</span></span></span></span></span></span></span></span></span></p> <p>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.</p>
title	Operational Objecthood as Separable Sharing
topic	Philosophy Mathematical logic
url	https://doi.org/10.5281/zenodo.20069437