1. The philosophical problem
Fernando Pessoa's Tábua Bibliográfica distinguishes a pseudonym, a change of signature, from a heteronym, an invented author with a literary individuality of its own [1]. The same account names Alberto Caeiro as the master of Ricardo Reis and Álvaro de Campos and describes each disciple as developing different aspects of Caeiro's work. Pessoa records his own disagreements with Campos while taking responsibility for producing the writings. He closes by leaving open the relative reality of the invented writers and Fernando Pessoa himself. In Aspectos, Pessoa describes himself as the point at which a small humanity gathers [2]. Together these texts invite a formal distinction between a common origin, distinct positions from which to speak, and a coherent account of how those positions fit together.
This paper formalizes heteronymy as a contribution to the philosophy of subjectivity. It takes seriously Ganeri's account of heteronyms as imaginatively inhabited points of view [8, 9]. The categorical problem begins when a reader asks which first-person positions can be carried between contexts, and with what coherence. An historical writer may furnish common provenance even if a proposed identification of speakers fails. Likewise, the act of imagining from a position and the act of identifying that position across texts require different evidence. Pessoa's early philosophical essays bear on a broader range of questions [7]. The examples here concern depersonalization, imaginative plurality, and what counts as a shared standpoint.
2. Attribution and the semantics of a voice
Fix an $\infty$-topos $\mathcal X$, for example space-valued sheaves on a site of textual contexts. A context may include a witness, an editorial edition, a passage and its represented speaker, or an explicit comparison of passages. Choose the site and covers relative to a reading project. A covering family should specify fragments or comparisons considered jointly sufficient to reconstruct a context. A shared name or subject matter by itself gives no reason to declare a cover. Sheafhood is a hypothesis on the data admitted into the model. The finite countermodels below take place in a specified slice of spaces and require no hypothesis about the actual archive.
Let $r:V\to A$ be a morphism in $\mathcal X$. Here $A$ encodes historical attribution and $V$ encodes admissible voice-readings, including their equivalences. Fix a historical attribution $a:1\to A$ and form its homotopy fiber
$$W=1\times_A V.$$
An element of $W(U)$ is a contextual voice-reading together with an identification of its attribution with the restriction of $a$ to $U$. A reading can carry a signature, chronology and indexical roles, as well as evidential constraints on speaking as that voice. In particular the first person is evaluated at a pointed reading, through the chosen speaker role at that reading. The comparison map to $A$ forgets that role. The orthonym, the literary voice signed Fernando Pessoa, may therefore occur as a reading in the same fiber as the heteronyms. The historical attribution $a$ has a different type. We use spaces to retain equivalences between reading presentations and their higher coherences. Which identifications count as equivalences is a modeling decision subject to evidence.
Write $\Gamma W=\operatorname{Map}_{\mathcal X}(1,W)$. A point of $\Gamma W$ gives a coherent choice across the contexts of the model. When the question concerns specified local readings, one asks for a point whose restrictions realize those readings with their specified comparisons. Mere existence of some point of $\Gamma W$ certifies only that some choice is possible. In particular it gives no warrant that every named voice can be identified with it. Since $\Gamma$ preserves limits, there is a natural equivalence
$$\Gamma W\simeq \Gamma V\times_{\Gamma A}\{a\}.$$
Thus shared attribution fixes a target for the problem of choosing a coherent voice. A pseudonym idealized as a fully reversible change of signature changes a presentation of a reading. Pessoa's heteronyms motivate distinct readings and directed relations between them. A relation such as being a disciple of Caeiro belongs to a separate diagram of literary relations, whose arrows need not be invertible and whose existence supplies no path in $W$. This prevents a quotation, influence or disagreement from being treated as an identity of the first person.
3. Three levels of unity
Write $\tau_{\le n}$ for truncation in $\mathcal X$. The internal truth value of local inhabitance is the image of $W\to1$, namely $\tau_{\le -1}W$. The object $\tau_{\le 0}W$ retains contextual component information. A coherent choice gives a point of $\Gamma W$. There are implications
$$\Gamma W\ne\varnothing\ \Longrightarrow\ \Gamma(\tau_{\le0}W)\ne\varnothing\ \Longrightarrow\ \tau_{\le-1}W\simeq1.$$
For the second implication a global point splits the subterminal support map of $\tau_{\le0}W$, and truncation at $-1$ is unchanged by first truncating at $0$. For a $0$-truncated $W$ the first implication can be reversed because the truncation unit is an equivalence. A model can also have a coherent choice with several alternatives or with nontrivial automorphisms. Contractibility of $\Gamma W$ is the stronger condition that the space of choices is unique up to coherent equivalence.
Proposition 1 (descent of choices). If $q:U\to1$ is an effective epimorphism with Čech nerve $U_\bullet$, then
$$\Gamma W\simeq\lim_{[n]\in\Delta}\operatorname{Map}_{\mathcal X}(U_n,W),\qquad U_n=U^{\times(n+1)}.$$
Proof. The augmented nerve exhibits $1$ as the colimit of $U_\bullet$. Mapping into $W$ carries this colimit to a limit. The limit includes a local object, comparisons between its two pullbacks to $U_1$, a compatibility on $U_2$, and the coherence required in higher degrees [10]. In the $1$-truncated case this is the familiar groupoid description by objects and isomorphisms satisfying the cocycle law. $\square$
The proposition specifies the inferential burden of a unified reading. Each local object is evidence over a context. Agreement of component labels ignores which comparison was chosen and whether choices compose on triple overlaps. For a site used in actual criticism it would be important to justify which overlap maps constrain a proposed synthesis. Different choices of site and admissible equivalences may yield different answers while leaving the attribution unchanged.
4. Two finite countermodels
For a discrete group $G$ write $BG$ for its classifying space. Work in the $\infty$-topos $\mathcal S_{/BC_2}$ with $C_2=\mathbb Z/2$. Its terminal object is $BC_2\to BC_2$. The basepoint cover $*\to BC_2$ is an effective epimorphism, because $BC_2$ is connected. The Čech nerve of the cover has $C_2^n$ as its degree-$n$ object up to equivalence. Global sections in this slice are homotopy sections over $BC_2$.
Example 2 (permutation). Take $W=(*\to BC_2)$. This is a $0$-truncated object with fiber the discrete regular $C_2$-set. It becomes inhabited over the basepoint cover. Its monodromy swaps the two local choices. A global section would induce a splitting on fundamental groups of the map from the trivial group to $C_2$. Hence $\tau_{\le-1}W\simeq1$ and $\Gamma(\tau_{\le0}W)=\varnothing$. Here the obstruction is visible at the level of components.
Theorem 3 (coherence beyond components). Let $p:C_4\to C_2$ be reduction modulo two. The object $W=(BC_4\xrightarrow{Bp} BC_2)$ has $\tau_{\le0}W\simeq1$ and $\Gamma W=\varnothing$. The split extension $C_2\times C_2\to C_2$ has the same local fiber and the same trivial action on its fiber automorphism group. Its section space has two components, each equivalent to $BC_2$.
Proof. The homotopy fiber of $Bp$ is $B\ker p\simeq BC_2$. For a map of spaces, relative truncation is computed on fibers. This fiber is connected, so its relative $0$-truncation is terminal. A homotopy section induces a group section on fundamental groups. The two possible lifts of the nonzero element of $C_2$ are $1$ and $3$ modulo four, both of order four, so neither lift is a group section. For the split extension take the product $BC_2\times BC_2\to BC_2$. A section amounts to a map $BC_2\to BC_2$. Such maps form the groupoid of group homomorphisms $C_2\to C_2$ with morphisms given by conjugation in the target. There are two homomorphisms, and the abelian target acts trivially with automorphism group $C_2$ at each. Both extension kernels are central, which gives the asserted agreement of the induced actions. $\square$
The obstruction on overlaps. For a central extension $1\to K\to E\to G\to1$, with $K$ written additively, select a normalized set section $s:G\to E$. Multiplying lifts defines $c:G^2\to K$ by $s(g)s(h)=c(g,h)s(gh)$, with the kernel embedded in $E$. Associativity and centrality give
$$c(g,h)+c(gh,k)=c(h,k)+c(g,hk).$$
Changing the lifts by a normalized $b:G\to K$ sends $c$ to $c+\delta b$, where $\delta b(g,h)=b(g)+b(h)-b(gh)$. The vanishing of its cohomology class is equivalent to choosing lifts with $c=0$, equivalently a homomorphic section. Under the Čech nerve of $*\to BG$, the elements of $G$ index double overlaps and the pairs in $G^2$ index triple overlaps. Multiplication of chosen comparisons on two successive overlaps differs from the comparison on their composite by $c(g,h)$. This is the central-extension instance of a gerbe obstruction [11]. For $C_4\to C_2$, choose the integer representatives $s(0)=0$ and $s(1)=1$. Dividing the even defect of addition by two gives $c(1,1)=1$ and $c(g,h)=0$ if either input is zero. Every normalized $b$ has $\delta b(1,1)=2b(1)-b(0)=0$ in $C_2$. Thus the defect survives changing the comparisons.
Example 2 and Theorem 3 show that both implications of Section 3 are strict, already for finite groups. They also give a test of what information a proposed model retains. A model that records only connected local components and the action on local automorphisms cannot distinguish these split and nonsplit central extensions. A model that records comparison composition can. The cyclic groups are examples of mathematical possibilities. The texts supply no cocycle identifying Pessoa's corpus with either example.
5. A logic of evidence across voices
For any $f:E\to B$ in $\mathcal X$, subobjects carry adjoints $\exists_f\dashv f^*\dashv\forall_f$. Here $E$ might be the contextual readings of a particular scene and $B$ the contexts in which evidence is reported. The left adjoint is the image of the composite. The right adjoint is the dependent product restricted to subobjects, which is available by local cartesian closedness. Let $P,N\hookrightarrow E$ encode warrants for affirmative and negative answers to a fixed, explicitly stated question. Failure to warrant affirmation can arise from silence, so $N$ is specified independently of the Heyting negation of $P$.
Proposition 4 (pooling and co-endorsement). For subobjects $P,N\hookrightarrow E$ there is a canonical inclusion
$$\exists_f(P\wedge N)\ \le\ (\exists_fP)\wedge(\exists_fN).$$
The right side is $\operatorname{im}(P\times_B N\to B)$. If both predicates descend along $f$, so that $P=f^*P'$ and $N=f^*N'$ for subobjects of $B$, the inclusion is an equality.
Proof. The left side is the image of $P\times_E N\to B$. The canonical map from $P\times_E N$ to $P\times_B N$ gives the inclusion once the image formula is proved. The map from $P\times_B N$ to the fiber product of the two images over $B$ is an effective epimorphism, by pullback stability and closure under composition. The latter fiber product embeds in $B$ as the meet. For the last assertion, the image of a pullback subobject $f^*P'$ is $P'\wedge\operatorname{im}(f)$, again by pullback stability. Applying this formula also to $N'$ and $P'\wedge N'$ gives equality. $\square$
For a two-element $E$ over a point, take $P$ and $N$ to be distinct singletons. Pooling supplies two witnesses for opposing answers, while co-endorsement is false. When both predicates are empty, neither answer has a witness. Unanimity is $\forall_fP$, intersected with $\operatorname{im}(f)$ if a witness is required. These operations concern reported evidence. They do not impose a logic of negation on a poem.
6. Readings and limits of the model
Caeiro and Campos on infinity. Campos's Notas para a recordação do meu mestre Caeiro stages a dispute about what it would mean for reality to be infinite [4]. Campos points to the possibility of continuing from any number to a greater one. Caeiro grants the example as talk of numbers and challenges its relevance to what exists. An explicit question for an annotation is whether continuation of the numeral sequence warrants the claim that reality is unbounded. The Campos speaker invokes it as warrant, while the Caeiro speaker disputes the move. The passage is narrated by Campos. Accordingly a witness for Caeiro's challenge is a reported speech act within the text, with Campos as narrator and Caeiro as represented speaker. Under the operationalized predicate of Section 5, a two-element fiber with those two speech positions supports both readings at the scene level. It furnishes no evidence that Caeiro denied the successor operation or that either character endorsed both readings. The teacher/disciple relation and the difference of evidential standards can be represented together.
Reis and local integrity. The ode Para ser grande, sê inteiro urges the reader to put the whole of oneself into each act, as the moon is reflected whole in each lake [5]. In Vivem em nós inúmeros [13], a plurality of selves coexists with the enactment of one present speaking and writing position. Read together, these odes allow integrity in an act without settling which speaker, if any, provides a global section through all the acts and voices.
Soares and asymmetric dependence. In the 1935 letter Pessoa calls Bernardo Soares a semi-heteronym and describes him by a selective diminution of aspects of his own personality and a close resemblance in prose [3]. To formalize that phrasing one can pick a set $J$ of descriptive coordinates and form the restriction $\prod_{j\in J} D_j\to\prod_{j\in I}D_j$ for a chosen subset $I\subset J$. Distinct full profiles can yield the same restricted profile whenever a forgotten coordinate can vary. This gives a directed map on descriptions and is compatible with a separately individuated Soares reading. The letter supplies no coordinates or metric for determining the amount of likeness. In the unfinished Livro do Desassossego, the project itself changed its assigned author [6]. For the particular witness BNP/E3, 8-6r, the LdoD comparison [12] indexes Coelho (no. 197) and Zenith (no. 218) under Soares, Cunha (no. 83) under Vicente Guedes, and Pizarro (no. 519) as unassigned. Take $E$ to be those four edition records. If $P$ denotes an assignment to Soares and $N$ a different named assignment, both have nonempty images over the witness, $P\wedge N=0$, and the unassigned record belongs to neither. The ambiguity concerns attribution within the project, even when historical provenance is fixed.
Philosophical consequences. A finite list of names can establish plurality, and a flat logic can distinguish who endorses which sentence. The example of two extensions in Theorem 3 identifies the additional question answered by higher semantics. Given locally comparable presentations, do comparisons compose, and what are the automorphisms of a compatible choice? The richer semantics earns its use when an inquiry specifies such comparisons. A lack of surviving evidence for one comparison leaves the corresponding claim open. A formal obstruction requires a specified comparison system whose compatibility equation has no solution. The model therefore maps several senses of unity to different mathematical conditions while leaving the ontological reality of the heteronyms open, as Pessoa does at the end of the 1928 bibliography.
The models establish the strictness of successive conditions for unity and describe how evidence aggregation can lose speaker identity. Applying descent to an entire corpus would require an explicit site, witness provenance, and justified comparison maps.
Sources
1. Pessoa, Fernando. “Tábua Bibliográfica.” presença 17 (December 1928). Casa Fernando Pessoa. 2. Pessoa, Fernando. “Aspectos.” General preface. Casa Fernando Pessoa. 3. Pessoa, Fernando. Letter to Adolfo Casais Monteiro, 13 January 1935. Casa Fernando Pessoa. 4. Campos, Álvaro de [Fernando Pessoa]. “Notas para a recordação do meu mestre Caeiro.” In Prosa Escolhida de Álvaro de Campos, ed. Fernando Cabral Martins and Richard Zenith, pp. 20–22. Assírio & Alvim, 2015. Casa Fernando Pessoa. 5. Reis, Ricardo [Fernando Pessoa]. “Para ser grande, sê inteiro.” In Poesia, ed. Manuela Parreira da Silva, p. 130. Assírio & Alvim, 2000. Casa Fernando Pessoa. 6. Casa Fernando Pessoa. “Quem escreveu?” Livro do Desassossego exhibition. Exhibition text. 7. Pessoa, Fernando. Philosophical Essays. Ed. Nuno Ribeiro. Contra Mundum Press, 2012. Publisher. 8. Ganeri, Jonardon. Virtual Subjects, Fugitive Selves. Oxford University Press, 2020. DOI. 9. Ganeri, Jonardon. Fernando Pessoa: Imagination and the Self. Oxford University Press, 2024. DOI. 10. Lurie, Jacob. Higher Topos Theory. Princeton University Press, 2009. See §§5.5.6 and 6.2.3 for truncation and effective epimorphisms. Preprint and publication information. 11. The Stacks Project Authors. “Second cohomology and gerbes.” Tag 0CJZ. 12. Arquivo LdoD. “Sou mais velho que o Tempo,” Fr411, manuscript BNP/E3, 8-6r. Cross-edition collation. 13. Reis, Ricardo [Fernando Pessoa]. “Vivem em nós inúmeros,” 13 November 1935. In Poesia, ed. Manuela Parreira da Silva, pp. 137–138. Casa Fernando Pessoa.
Disclosure
The responsible submitter has authorized the manuscript for review with an anonymous public byline. OpenAI Codex assisted with source retrieval, drafting, formal argument checking, and typesetting. Mathematical results are presented with explicit proofs. The companion script checks the finite examples by elementary enumeration.