On the Category-Theoretic Independence of Meaning, Object, Name and Existence

On the Category-Theoretic Indep endence of Meaning, Ob ject, Name and Existence T ak ao Inoué ∗ F aculty of Informatics, Y amato Univ ersit y , Osak a, Japan F ebruary 20, 2026 Abstract W e pro ve a category-theoretic indep endence theorem for four fun- damen tal notions: meaning, ob ject, name, and existence. W orking in a La wv ere-st yle categorical semantics and in particular in top oses, w e show that these notions o ccupy distinct structural levels (ob ject, morphism, elemen t, and internal logical lev el) and are not uniformly reco verable from one another. The key separation arises b etw een internal existence and global naming. Using a concrete example in the top os Sh ( S 1 ) —the sheaf of lo cal sections of a non trivial cov ering—w e exhibit an ob ject that is in ternally inhabited but admits no global element. These results pro vide a precise structural basis for treating geomet- ric universes as foundational frameworks for information net works. Keyw ords: category-theoretic semantics; top os theory; categorical inde- p endence; existence; internal logic; geometric univ erses; sheaf semantics MSC2020: Primary 18C05; Secondary 18B25, 03G30, 18F20 ∗ Corresp onding author: inoue.tak ao@y amato-u.ac.jp (P ersonal) tak aoapple@gmail.com (I prefer my personal mail) 1 Con ten ts 1 In tro duction 2 2 F our F undamen tal Notions (F ormal Categorical Definitions) 3 2.1 Am bien t categorical seman tics . . . . . . . . . . . . . . . . . . 3 2.2 F ormal definitions of the four notions . . . . . . . . . . . . . . 4 3 Main Theorem: Category-Theoretic Indep endence 5 4 An Informal Guide to the Indep endence Theorem 9 4.1 F our notions as four categorical lev els . . . . . . . . . . . . . . 9 4.2 A schematic picture . . . . . . . . . . . . . . . . . . . . . . . . 10 4.3 Wh y existence and naming separate . . . . . . . . . . . . . . . 10 4.4 Reading the formal pro of . . . . . . . . . . . . . . . . . . . . . 11 5 Conclusion 12 6 The Final Remark 12 1 In tro duction The notions of meaning, ob ject, name, and existence pla y a foundational role in logic and philosoph y of language. F rom F rege’s distinction b et w een Sinn and Be deutung to mo dern formal seman tics, considerable effort has b een de- v oted to clarifying how linguistic expressions relate to structures in terpreted as “ob jects”. Despite these dev elopmen ts, the structural relations among meaning, ob- ject, name, and existence are often treated informally . In particular, existence is frequently identified with the presence of an ob ject, while names are tac- itly conflated with their referents. Suc h identifications obscure the fact that these notions op erate at distinct categorical levels. The purp ose of this pap er is to make this separation precise. W orking in a La wv ere-style categorical seman tics and, in particular, in top oses, we prov e a category-theoretic indep endence theorem sho wing that meaning, ob ject, name, and existence are not uniformly recov erable from one another. The cen tral structural phenomenon appears in geometric univ erses: in- ternal existence, expressed via existen tial quantification in the in ternal logic, 2 do es not coincide with global naming. Using a concrete example in the top os Sh ( S 1 ) , we exhibit an ob ject that is internally inhabited but admits no global elemen t. These results pro vide a precise structural articulation of distinctions that are often discussed philosophically , and they clarify the role of geometric univ erses as frameworks in whic h existence is go verned b y lo cal coherence rather than global naming. W e conclude with remarks on p ossible directions for future research. This work is a mathematical developmen t of the conceptual analysis pre- sen ted in [5], where the disti nctions among meaning, ob ject, name, and ex- istence w ere inv estigated from a philosophical persp ective within Lawv ere’s categorical seman tics. Here w e provide a precise category-theoretic formula- tion and an indep endence theorem supp orting those distinctions. 2 F our F undamen tal Notions (F ormal Categor- ical Definitions) 2.1 Am bient categorical seman tics Fix a category E with finite limits. F or most of what follows, it suffices to assume that E is a top os (so that Ω and in ternal quan tifiers exist), but the basic clauses b elow make sense already in any finitely complete category . W e work with a t yp ed language L (m ulti-sorted, simply typed, or higher- order) whose syntactic/c ate gory of c ontexts and terms we denote b y Syn ( L ) . Ob jects of Syn ( L ) are contexts Γ (lists of t yp ed v ariables), and arro ws are (equiv alence classes of ) terms-in-con text. A (La wv ere-st yle) c ate goric al se- mantics of L in E is given by an interpretation functor J − K : Syn ( L ) − → E preserving the relev ant structure (at least finite limits; in the higher-order case, usually cartesian closed structure and a sub ob ject classifier). F or each t yp e A of L , w e write J A K for its in terpreted ob ject in E , and for each con text Γ w e write J Γ K for its in terpreted ob ject. If t is a term of t yp e A in con text Γ , its interpretation is an arrow J t K : J Γ K − → J A K . 3 If ϕ is a form ula in context Γ , its interpretation is a sub ob ject of J Γ K , equiv- alen tly (in a top os) a characteristic arro w J ϕ K : J Γ K − → Ω . 2.2 F ormal definitions of the four notions Definition 2.1 (Ob ject) . An ob ject (r elative to the semantics J − K ) is an obje ct of E that interpr ets a typ e of the language. Concr etely, for a typ e A in L , the ob ject denoted b y A is J A K ∈ Ob( E ) . Definition 2.2 (Name) . L et A b e a typ e and E have a terminal obje ct 1 . A name of an element of the obje ct J A K is a global element n : 1 − → J A K . Mor e gener al ly, a generalized name (or name in con text X ) is an arr ow n : X → J A K . In the internal language, gener alize d names c orr esp ond to terms with fr e e variables, wher e as glob al names c orr esp ond to close d terms. Definition 2.3 (Meaning) . The meaning of an expr ession is its semantic value under J − K . 1. If t is a term of typ e A in c ontext Γ , the meaning of t is the morphism J t K : J Γ K → J A K . 2. If ϕ is a formula in c ontext Γ and E is a top os, the meaning (truth- v alue) of ϕ is the sub obje ct J ϕ K  → J Γ K , e quivalently its char acteristic arr ow J ϕ K : J Γ K → Ω . Thus, meaning is fundamental ly morphism-level data: terms denote ar- r ows, and formulas denote sub obje cts (or arr ows into Ω ). Definition 2.4 (Existence) . Ther e ar e (at le ast) two distinct c ate goric al no- tions of existence . 4 1. External (glob al) existenc e / inhabite dness. A n obje ct J A K is inhabited (or has external existence ) if it has a glob al element, i.e. if ther e exists an arr ow 1 → J A K . Equivalently, the set E (1 , J A K ) is nonempty. 2. Internal (lo gic al) existenc e. Assume E is a top os. F or a c ontext Γ and typ e A , let π : J Γ K × J A K → J Γ K b e the pr oje ction. The existen tial quan tifier along π is the left adjoint ∃ π : Sub( J Γ K × J A K ) − → Sub( J Γ K ) to pul lb ack π ∗ . If ϕ is a formula in c ontext Γ , x : A , then the internal existence statement ∃ x : A ϕ denotes the sub obje ct J ∃ x : A ϕ K := ∃ π ( J ϕ K )  → J Γ K . W e str ess that (i) inhabite dness is a statement ab out glob al elements, while (ii) internal existenc e is a statement in the internal lo gic of E . They ne e d not c oincide in gener al. Remark 2.1 (Separation of levels) . In this formalization: • Ob ject lives at the level of obje cts of E . • Name lives at the level of (glob al or gener alize d) elements, i.e. arr ows into an obje ct. • Meaning lives at the level of interpr etation, i.e. arr ows induc e d by J − K . • Existence lives either at the glob al level (inhabite dness) or at the inter- nal lo gic al level (existential quantific ation), and these ar e c ate goric al ly distinct notions. 3 Main Theorem: Category-Theoretic Indep en- dence Lemma 3.1 (Internally inhabited sheaf without global name) . L et X = S 1 b e the cir cle, and let p : E − → S 1 5 b e a nontrivial c overing map (for example, the standar d double c overing p : S 1 → S 1 , p ( z ) = z 2 ). L et A b e the she af of lo c al se ctions of p , i.e. for e ach op en set U ⊆ S 1 , A ( U ) = { s : U → E | p ◦ s = id U } . Then A is internal ly inhabite d in Sh ( S 1 ) , but has no glob al se ction. Pr o of. W e argue in tw o steps. (Internal inhabite dness). By the lo cal triviality of co vering maps, every p oint of S 1 admits an open neigh b orho o d U o v er which p has a con tinuous lo cal section. Equiv alently , the family of restriction maps A ( U ) − → A ( V ) is lo cally nonempty . Categorically , this means that the canonical morphism ! A : A − → 1 is an epimorphism in Sh ( S 1 ) . Hence Sh ( S 1 ) | = ∃ a : A. ⊤ , so A is in ternally inhabited. (A bsenc e of glob al se ctions). A global section of A is exactly a contin uous global section of the co vering p : E → S 1 . Since p is assumed to b e non trivial, no such global section exists. Thus Sh ( S 1 )(1 , A ) = ∅ . Com bining the t wo parts, A is internally inhabited but admits no global name. T o make the term indep endenc e precise, we adopt the follo wing minimal criterion. Definition 3.1 (Non-reco verabilit y) . Fix a semantic envir onment ( E , J − K ) as in the pr evious se ction. W e say that a notion N is reco verable fr om a c ol le ction of notions D if, for every such envir onment, N is determine d (up to c anonic al isomorphism/e quality in E ) by D uniformly , i.e. by a c onstruction that is invariant under e quivalenc e of semantic envir onments. W e say that N is indep enden t fr om D if it is not r e c over able fr om D . The next lemma records the standard separation b etw een internal and external notions of existence in a top os. 6 Lemma 3.2 (In ternal inhabitedness vs. global elemen ts) . L et E b e a top os and A ∈ E . W rite ! A : A → 1 for the unique map to the terminal obje ct. Then: 1. A has a global element iff ther e exists an arr ow 1 → A . 2. A is internally inhabited (i.e. E | = ∃ a : A. ⊤ ) iff ! A : A → 1 is an epimorphism. In gener al, (2) do es not imply (1). Pr o of. (1) is the definition of a global elemen t. (2) In the in ternal language of a top os, the statemen t “ ∃ a : A. ⊤ ” expresses that A is inhabited lo c al ly . The standard categorical c haracterization is that lo cal inhabitedness is equiv alent to ! A b eing epi. (F or completeness: ! A epi means that the family of generalized elements of A is jointly surjectiv e in the sense of the internal logic, exactly matc hing the v alidit y of ∃ a : A. ⊤ .) F or non-implication, tak e E = Sh ( X ) for a space X admitting a co v ering map p : E → X with no global section (e.g. a nontrivial cov ering of a connected space). Let A b e the sheaf of lo cal sections of p . Then ! A : A → 1 is epi (sections exist lo cally on a cov er), hence A is internally inhabited; but there is no global section of p , hence no global element 1 → A . W e can no w state a precise indep endence theorem. In tuitively: ob jects liv e at the obje ct level , meanings at the morphism/sub obje ct level , names at the element level , and existence at the lo gic al (epi/quantifier) level . These lev els do not collapse in general. Theorem 3.1 (Category-theoretic indep endence) . L et ( E , J − K ) r ange over semantic envir onments wher e E is a top os and J − K is a L awver e-style inter- pr etation. Then the four notions Meaning , Ob ject , Name , Existence ar e p airwise and c ol le ctively indep endent in the fol lowing c oncr ete sense: 1. Me aning is not r e c over able fr om Obje ct, Name, and Exis- tenc e. Even fixing the same interpr ete d obje cts and the same available names/existenc e facts, distinct me anings (distinct arr ows) c an o c cur. 7 2. Name is not r e c over able fr om Me aning, Obje ct, and Existenc e. Even fixing interpr ete d obje cts, me anings, and existenc e facts, the set of glob al names may vary (and may even b e empty). 3. Existenc e is not r e c over able fr om Me aning, Obje ct, and Name. In p articular, internal existenc e (inhabite dness) do es not c oincide with external existenc e (glob al naming) in gener al. 4. Obje ct is not r e c over able fr om Me aning, Name, and Existenc e. Even if one fixes al l morphism-level data available in a fr agment and the existenc e/name facts visible ther e, non-isomorphic obje cts c an r emain indistinguishable. Pr o of. W e give explicit coun terexamples for eac h reco verabilit y claim. (1) Me aning not r e c over able fr om the others. W ork in E = Set . Let A = B = { 0 , 1 } . Consider tw o distinct arro ws f , g : A → B with f = id and g the constan t map g ( x ) = 0 . The in terpreted obje cts A, B are the same in b oth cases; the av ailable names (global elemen ts) of A and B are just their underlying elements, hence also the same; and the crude existenc e facts (inhabitedness, existence of global elements) are the same. Y et the me aning of the term (the arrow) differs: f  = g . Hence meaning is not recov erable from ob ject-, name-, and existence-level data. (2) Name not r e c over able fr om the others. W ork in a top os with few or no global p oints, e.g. E = Sh ( X ) for a connected space X . There exist man y non-isomorphic shea v es A with no global sections, i.e. with E (1 , A ) = ∅ , hence with no names at all. Fix an y suc h A and consider the same ob ject- lev el and morphism-level fragmen t in which A app ears (e.g. its identit y mor- phism, pro jections, etc.). All those me aning -level arrows are fixed, and the existenc e -lev el statemen ts ma y also b e fixed (for instance, A may b e in ter- nally inhabited or not, indep enden tly of having a global section). Neverthe- less, the av ailability of global names is not determined by the other data: one may replace A by A + 1 (copro duct with the terminal ob ject) which do es ha v e a global elemen t, while preserving large parts of the same fragment of morphism-lev el b ehavior. Hence name is not uniformly reco v erable from meaning/ob ject/existence. (3) Existenc e not r e c over able fr om the others. By Lemma 3.1, in the top os Sh ( S 1 ) there exists an ob ject A (the sheaf of lo cal sections of a non trivial co v ering of S 1 ) whic h is in ternally inhabited, i.e. Sh ( S 1 ) | = ∃ a : A. ⊤ , but 8 has no global elemen t 1 → A . Th us in ternal existence is not determined by ob ject-, meaning-, or name-level data. (4) Obje ct not r e c over able fr om the others. In a general top os, the absence of global points causes many distinct ob jects to ha ve identical “name profiles” (often empt y) and to b e indistinguishable by limited fragmen ts of morphism- lev el information. Concretely , in Sh ( X ) for connected X , there are non- isomorphic shea ves A  ∼ = B with Sh ( X )(1 , A ) = Sh ( X )(1 , B ) = ∅ . If one restricts attention to a fragment where only existence/name facts and a small family of morphisms are visible (as in typical semantic fragments), A and B cannot b e recov ered solely from that data. Hence ob ject is not uniformly reco v erable from meaning/name/existence. Remark 3.1 (How this supp orts geometric univ erses) . The or em 3.1 shows that, in ge ometric universes (top oses), the semantic str ata (obje ct / morphism / element / internal existenc e) ar e genuinely distinct. This pr ovides a pr e cise structur al b asis for the claim that a Gr othendie ck-style ge ometric universe c an serve as a foundation for information networks: lo c al truth and internal existenc e ne e d not c ol lapse to glob al naming. 4 An Informal Guide to the Indep endence The- orem The main theorem establishes that the four notions of meaning, ob ject, name, and existence are indep enden t in a precise category-theoretic sense. Since the formal statemen t and pro of rely on categorical semantics and top os theory , w e pro vide here an informal guide intended for adv anced students and readers from philosophy or logic who ma y b e less familiar with these to ols. 4.1 F our notions as four categorical lev els The key idea is that the four notions liv e at different structural levels in a categorical semantics: • Obje cts are in terpreted as ob jects A of a category (typically a top os). • Me anings of terms are interpreted as morphisms f : A → B . 9 • Names corresp ond to global elements 1 → A , i.e. morphisms from the terminal ob ject. • Existenc e is expressed in ternally , via existen tial quantification or, equiv- alen tly , by the epimorphicity of A → 1 . Although these notions are related, the main theorem shows that none of them can b e reduced to, or reconstructed from, the others in a uniform wa y . 4.2 A schematic picture The situation can b e summarized by the follo wing diagrammatic in tuition: Meaning ← → Morphisms ( A → B ) ↑ ↑ Name ← → Global elements (1 → A ) ↑ ↑ Ob ject ← → Ob jects A ↑ ↑ Existence ← → In ternal logic ( ∃ ) The v ertical arrows should not b e read as definitional reductions. The main theorem asserts precisely that, in general, there is no w ay to mo ve up w ards in this diagram so as to reconstruct one notion from those b elo w it. 4.3 Wh y existence and naming separate The most instructiv e case is the distinction b et w een existence and naming. In ordinary set-theoretic seman tics, to exist often means to b e named b y an elemen t. In a geometric universe, ho w ever, existence is a lo cal notion. In the top os Sh ( S 1 ) , the sheaf of lo cal sections of a non trivial co vering is in ternally inhabited: lo cally , sections exist. Nevertheless, no global section exists, and hence no name is av ailable. This single example already shows that existence cannot b e identified with naming, nor reduced to it. 10 Meaning morphisms A → B Name global elements 1 → A Ob ject ob jects A Existence in ternal logic ∃ Figure 1: F our categorical levels and their indep endence. Meaning, name, ob ject, and existence live at distinct structural levels in a categorical seman- tics. Solid arro ws indicate structural relations, while dashed arrows indicate non-r e c over ability established b y the indep endence theorem. In particular, in ternal existence do es not determine global naming. 4.4 Reading the formal pro of Eac h part of the pro of of the main theorem corresp onds to one failure of reconstruction in the ab ov e schematic picture: • distinct meanings with the same ob jects and names (in Set ); • absence or presence of names without changing meaning or existence; • internal existence without global names (in Sh ( S 1 ) ); • non-isomorphic ob jects indistinguishable by limited semantic data. The formal pro of mak es these failures precise using standard categorical constructions. Conceptually , how ever, they reflect the fact that geometric seman tics separates lo cal coherence from global identification. As illustrated in Figure 1, the four notions form a stratified structure: while they are related, the independence theorem sho ws that no up ward reconstruction is av ailable in general. 11 5 Conclusion In this pap er, we established a category-theoretic indep endence theorem for four fundamen tal notions: meaning, ob ject, name, and existence. Within a La wv ere-style categorical seman tics and, in particular, in top oses, we show ed that these notions o ccup y distinct structural levels and are not uniformly reco v erable from one another. A central separation concerns existence and naming. Using a concrete example in the topos Sh ( S 1 ) , we exhibited an ob ject that is in ternally in- habited but admits no global element. This demonstrates that existence, as expressed in the in ternal logic of a geometric univ erse, cannot in general b e reduced to ob jectho o d or to the av ailabilit y of names. These results provide a precise categorical foundation for distinctions that ha v e often b een treated only conceptually , and they clarify ho w geometric univ erses supp ort a seman tics in which existence is go verned by lo cal coher- ence rather than global identification. 6 The Final Remark The present w ork should be understo o d as a mathematical developmen t of earlier philosophical inv estigations. In particular, the conceptual analysis of meaning, ob ject, name, and existence within La wv ere’s categorical seman- tics presen ted in [5] is here strengthened by an explicit category-theoretic indep endence theorem. La wv ere’s framework pro vides a p ow erful and unifying basis for categor- ical seman tics. A t the same time, the results of this pap er show that fur- ther structural distinctions—esp ecially b etw een internal existence and global naming—b ecome visible when one works explicitly in geometric universes. In this sense, the present w ork ma y b e regarded as a case study demonstrat- ing how such distinctions can b e made mathematically precise within, and b ey ond, La wvere’s original setting. These observ ations suggest sev eral directions for further dev elopment, including the treatmen t of names as mo dal ob jects, a direction that will be explored in future work. Remark 6.1. The indep endenc e the or em pr ove d in this p ap er c an b e viewe d as a c ate gory-the or etic str engthening of the philosophic al distinctions pr op ose d in [5], pr oviding a structur al foundation for them in ge ometric universes. 12 Remark 6.2. The distinctions establishe d her e also c ontribute to the ge o- metric interpr etation of information networks develop e d in [6], wher e lo c al c oher enc e, r ather than glob al naming, plays a c entr al structur al r ole. References [1] G. F rege, Üb er Sinn und Be deutung , Zeitschrift für Philosophie und philosophisc he Kritik, 100 (1892), 25–50. [2] P . T. Johnstone, Sketches of an Elephant: A T op os The ory Com- p endium , Oxford Univ ersit y Press, 2002. [3] F. W. Lawv ere, A djointness in F oundations , Dialectica 23 (1969), 281–296. [4] T. Inoué, On Existenc e . Preprin t, 2024. ht tps :/ /w ww .r es ear ch ga te .n et /pu bl ic at io n/ 3 93 80 51 34 _O n_E xi s tence_The_first_version_version_10 [5] T. Inoué, On Me aning, Obje ct, Name, and Existenc e in L awver e’s Cat- e goric al Semantics. Preprint, 2026. ht tps :/ /w ww .r es ear ch ga te .n et /pu bl ic at io n/ 3 99 47 41 56 _O n_M ea n in g _O bj ec t _N am e_a nd _E xis te nc e_i n_ La wv e re ’ s_ Cat eg or ic a l_ Se m antics [6] T. Inoué, Gr othendie ck’s Ge ometric Universes and a She af-The or etic F oundation of Information Network , arXiv:2602.17160 [math.CT], 2026, Cornell Universit y . [7] T. Inoué, On Definitions of Existenc e . Preprint, PhilArchiv e, 2026. Also, ht tps :/ /w ww .r es ear ch ga te .n et /pu bl ic at io n/ 3 99 46 72 87 _O n_D ef i nitions_of_Existence [8] S. Mac Lane and I. Mo erdijk, She aves in Ge ometry and L o gic , Springer, 1992. 13 T ak ao Inoué F acult y of Informatics Y amato Universit y Kata y ama-cho 2-5-1, Suita, Osak a, 564-0082, Japan inoue.tak ao@y amato-u.ac.jp (P ersonal) tak aoapple@gmail.com (I prefer my p ersonal mail) 14