Semiotic Universe

Abstract

In this paper, we’ll construct a “semiotic universe”: a mathematical universe in which:

• semantic objects form a complete Heyting algebra, equipped with:
  • a modal closure operator $j$, and
  • a trace comonad $G$, that’s compatible with $j$.
• syntactic operators generated by a typed lambda calculus
• a fusion mechanism interprets those syntactic operators as semantic endomorphisms, enforcing coherence between syntax and semantics.

Three monotone, inflationary closure operators ${\mathcal{S}}_{\mathrm{sem}},{\mathcal{S}}_{\mathrm{syn}},{\mathcal{S}}_{\mathrm{fus}}$ act on the complete lattice of partial structures. Their composite $\mathcal{S}$ has a least fixed point, which we define as the semiotic universe.

We’ll prove that this semiotic universe is initial in a natural (2-)category of semiotic structures: it is the free modal–comonadic Heyting structure equipped with an interpretation of the given syntax.

Ambient Semantic and Algebraic Structure

To begin, let’s fix an ambient environment for our semantics.

1.1 Complete Heyting Algebra

A complete Heyting algebra is a complete lattice $(H,\le )$ equipped with:

• binary meet $\wedge$,
• binary join $\vee$,
• top $\top$,
• bottom $\perp$,
• implication $\Rightarrow$,

such that for all $a,b,c\in H$:

$$c\le (a\Rightarrow b)\text{iff}c\wedge a\le b.$$

We assume throughout:

• $H$ is a complete Heyting algebra;
• all semantic constructions are built in this algebra.

The underlying poset $(H,\le )$ is viewed as a thin category:

• objects: elements of $H$,
• a unique morphism $a\to b$ iff $a\le b$.

1.2 Modal Closure Operator

A modal closure operator on $H$ is a function $j:H\to H$ such that:

1. (Extensive) $a\le j(a)$;
2. (Monotone) $a\le b⟹j(a)\le j(b)$;
3. (Idempotent) $j(j(a))=j(a)$;
4. (Join-continuous) for any family $\{a_i{\}}_{i\in I}$,

$$j\left(\underset{i\in I}{\bigvee }{a}_i\right)=\underset{i\in I}{\bigvee }j(a_i).$$

The stable / modal fragment is

$$H^{\mathrm{st}}:=\{a\in H:j(a)=a\}.$$

Lemma 1.1 (Modal fragment is a Heyting subalgebra)

$H^{\mathrm{st}}$ is a complete Heyting subalgebra of $H$, and the inclusion $i:H^{\mathrm{st}}\hookrightarrow H$ is a Heyting homomorphism.

Sketch. Join-continuity of $j$, combined with the adjunction definition of $\Rightarrow$, ensures that $H^{\mathrm{st}}$ is closed under all Heyting operations, and that the inclusion preserves them.

1.3 Heyting–Comonadic Trace Operator

We now specify a trace operator that is both comonadic and algebraically compatible with the Heyting structure.

A comonad on the poset-category of $H$ is given by:

• a monotone endofunctor $G:H\to H$,
• a counit ${ϵ}_a:G(a)\le a$,
• a comultiplication ${\delta }_a:G(a)\le G(G(a))$,

natural in $a$, such that:

• $G({\delta }_a)\circ {\delta }_a\le {\delta }_{G(a)}\circ {\delta }_a$ (coassociativity),
• ${ϵ}_{G(a)}\circ {\delta }_a\le {\mathrm{id}}_{G(a)}$,
• $G({ϵ}_a)\circ {\delta }_a\le {\mathrm{id}}_{G(a)}$ (counit laws).

We further impose algebraic compatibility:

Axiom 1.2 (Heyting–comonadic coherence)

The trace $G:H\to H$ is a complete Heyting algebra endomorphism and join-continuous. That is, for all families $\{a_i\}$, and all $a,b\in H$,

1. $G({\bigvee }_i{a}_i)={\bigvee }_{i}G(a_i)$ (join-continuous);
2. $G(a\wedge b)=G(a)\wedge G(b)$, $G(\top )=\top$, $G(\perp )=\perp$;
3. $G(a\Rightarrow b)=G(a)\Rightarrow G(b)$.

So $G$ is simultaneously:

• a comonad on the poset-category underlying $H$, and
• a homomorphism of complete Heyting algebras.

This is the missing coherence law: it ensures that temporal unfolding respects the logical structure.

1.4 Modality–Trace Interaction and Stability

We require:

Axiom 1.3 (Basic interaction)

For all $a\in H$,

$$j(G(a))\le G(j(a)).$$

Axiom 1.4 (Stability with respect to trace)

For all $a\in H$,

1. if $a\in H^{\mathrm{st}}$ then $G(a)\in H^{\mathrm{st}}$;
2. if $G(a)\in H^{\mathrm{st}}$ then $a\in H^{\mathrm{st}}$.

Thus

$$G(a)\in H^{\mathrm{st}}⟺a\in H^{\mathrm{st}}.$$

Lemma 1.5

The restriction of $G$ to $H^{\mathrm{st}}$ is a comonad and a Heyting endomorphism on the modal fragment, and

$$(H^{\mathrm{st}},j=\mathrm{id},G{|}_{H^{\mathrm{st}}})$$

is a modal–temporal substructure of $(H,j,G)$.

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2. Syntactic Operator Algebra

We now describe the syntactic side at an abstract level.

2.1 Types, Terms, and Raw Operators

We fix:

• a type language $\mathsf{Ty}$, generated from
  • a base type $P$,
  • function types $A\to B$,
  • product types $A\times B$;
• a λ-calculus of terms with variables, λ-abstraction, application, pairing, projections, and chosen primitive constants;
• typing judgments $\Gamma ⊢t:A$, with standard rules and structural properties (subject reduction, weakening, exchange, substitution).

For each $n\ge 0$, define:

• ${\mathsf{Op}}_n$: the set of closed terms $f$ with type $P^n\to P$;
• ${\mathsf{Op}}_n/\equiv$: quotient by definitional equality (βη and any additional equational axioms).

Let

$${\mathsf{Op}}^{\mathrm{raw}}:=\bigcup _{n\ge 0}{\mathsf{Op}}_n/\equiv .$$

2.2 Definable Operators

Definition 2.1 (Definable operators)

For each $n$, ${\mathsf{Op}}_n^{\mathrm{def}}\subseteq {\mathsf{Op}}_n/\equiv$ is the smallest set that:

1. contains all primitive operator constants of arity $n$;
2. is closed under λ-abstraction and application yielding type $P^n\to P$;
3. is closed under composition of operators at type $P$.

Set

$${\mathsf{Op}}^{\mathrm{def}}:=\bigcup _{n\ge 0}{\mathsf{Op}}_n^{\mathrm{def}}.$$

We treat ${\mathsf{Op}}^{\mathrm{def}}$ as a finitary algebra freely generated by primitive operations under λ-definability and composition.

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3. Fragments and Restriction

We formalize the local “fragmentwise” reasoning used in fusion and compactness.

3.1 Modal–Temporal Heyting Subalgebras

Definition 3.1 (Modal–temporal subalgebra)

A subset $F\subseteq H$ is a modal–temporal subalgebra if:

1. $F$ is closed under $\wedge ,\vee ,\Rightarrow ,\perp ,\top$;
2. $F$ is closed under $j$;
3. $F$ is closed under $G$.

By Axiom 1.2, $G$ preserves all Heyting structure, so this notion is coherent.

Definition 3.2 (Fragments)

A fragment is a finitely generated modal–temporal subalgebra $F\subseteq H$: there exists a finite set $S\subseteq F$ such that:

• $F$ is the smallest modal–temporal subalgebra containing $S$.

Let $\mathcal{F}(H)$ be the set of fragments.

3.2 Restriction of Operators

Let $f:H^n\to H$ be a function.

• If $f(F^n)\subseteq F$, we define its restriction

$$f{|}_F:F^n\to F$$

as the obvious induced function.

• If not, then $f$ is not fragment-preserving on $F$.

Definition 3.3 (Fragment-preserving operators)

An operator $f:H^n\to H$ is fragment-preserving if

$$f(F^n)\subseteq F$$

for every fragment $F\in \mathcal{F}(H)$.

For fragment-preserving operators, restriction to fragments is always defined.

3.3 Fragmentwise Equality and Hereditary Extensionality

Definition 3.4 (Fragmentwise equality)

For fragment-preserving $f,g:H^n\to H$ and a fragment $F\in \mathcal{F}(H)$, we say

$$f{\equiv }_{F}g$$

if $f(a_1,\dots ,a_n)=g(a_1,\dots ,a_n)$ for all $a_i\in F$.

Definition 3.5 (Hereditarily extensional family)

A family $\mathcal{E}$ of fragment-preserving operators is hereditarily extensional if:

whenever $f,g\in \mathcal{E}$ and there is a fragment $F$ such that $f{\equiv }_{F}g$, then for any fragment $F^{\prime }$ obtained from $F$ by finitely many applications of:

• Heyting operations,
• the modality $j$,
• the trace $G$,
• any operator in $\mathcal{E}$, we have $f{\equiv }_{F^{\prime }}g$.

This captures the idea that equalities valid on a fragment remain valid under all semantic construction steps relevant to closure.

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4. Interpretation and Modal–Comonadic Operators

We now connect the syntactic algebra to the semantic structure.

4.1 Interpretation of Definable Operators

An interpretation is a family of maps

$$[[-{]]}_n:{\mathsf{Op}}_n^{\mathrm{def}}\to \mathrm{Hom}(H^n,H)$$

such that, for each $f\in {\mathsf{Op}}_n^{\mathrm{def}}$,

1. (Monotone & join-continuous) $[[f]]$ is monotone in each argument and preserves directed joins in each argument.
2. (Heyting compatibility) The operators corresponding to $\wedge ,\vee ,\Rightarrow ,\perp ,\top$ on $P$ are interpreted as the Heyting operations on $H$.
3. (Modal homomorphism)

$$j([[f]](a_1,\dots ,a_n))=[[f]](j(a_1),\dots ,j(a_n))\forall a_i\in H.$$

4. (Trace compatibility)

$$G([[f]](a_1,\dots ,a_n))=[[f]](G(a_1),\dots ,G(a_n))\forall a_i\in H.$$

5. (Definitional equality preservation) If the λ-theory proves $f\equiv g$, then

$$[[f]]=[[g]].$$

6. (Fragment preservation) For every fragment $F$, we have:

$$[[f]](F^n)\subseteq F.$$

7. (Hereditarily extensional family) The family $\{[[f]]:f\in {\mathsf{Op}}^{\mathrm{def}}\}$ is hereditarily extensional in the sense of Definition 3.5.

The fragment preservation axiom (6) explicitly guarantees that definable operators preserve fragments as algebras; we do not attempt to derive it from weaker assumptions. This directly resolves the fragment-preservation issue.

Proposition 4.1 (Modal fragment as type discipline)

For each definable operator $f$, the restriction

$$[[f]]:(H^{\mathrm{st}}{)}^n\to H^{\mathrm{st}}$$

is well-defined. Moreover, on the modal fragment, it commutes with $G$ and preserves the Heyting structure.

Sketch. If all $a_i\in H^{\mathrm{st}}$, then (3) implies their image is stable under $j$, so lies in $H^{\mathrm{st}}$. Axiom 1.2 and (4) ensure compatibility with Heyting structure and $G$. Fragment preservation (6) ensures that modal–temporal fragments remain closed under these operators.

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5. Semiotic Fusion: Congruence and Naming

Fusion enforces coherence between syntax and semantics by:

• quotienting syntactic operators by fragmentwise semantic equality,
• adding new operators that name already-available semantic behaviors.

5.1 Partial Semiotic Structures

A partial semiotic structure is a pair

$$X=(H_X,{\text{Op}}_X)$$

with:

• $H_X\subseteq H$ a set of semantic objects;
• ${\text{Op}}_X\subseteq {\mathsf{Op}}^{\mathrm{def}}$ a set of syntactic operators.

We form the complete lattice

$$\mathcal{U}:=\mathcal{P}(H)\times \mathcal{P}({\mathsf{Op}}^{\mathrm{def}})$$

ordered by $(H_X,{\text{Op}}_X)\le (H_Y,{\text{Op}}_Y)$ iff $H_X\subseteq H_Y$ and ${\text{Op}}_X\subseteq {\text{Op}}_Y$.

5.2 Fragmentwise Congruence

For a partial structure $X=(H_X,{\text{Op}}_X)$, let $\mathcal{F}(H_X)$ denote the set of fragments $F\in \mathcal{F}(H)$ with $F\subseteq H_X$.

Definition 5.1 (Congruence ${\sim }_X$)

For operators $f,g\in {\text{Op}}_X$, we write

$$f{\sim }_{X}g$$

if:

1. for every fragment $F\in \mathcal{F}(H_X)$,

$$[[f]]{\equiv }_F[[g]];$$

2. this equality is hereditarily extensional: closed under applying Heyting operations, $j$, $G$, and any operator in ${\text{Op}}_X$.

Because the interpretation family is hereditarily extensional, ${\sim }_X$ is a well-defined congruence relation.

5.3 Naming Existing Behaviors

For each partial structure $X=(H_X,{\text{Op}}_X)$, consider the closure (in the hereditarily extensional family) of $\{[[f]]:f\in {\text{Op}}_X\}$. Any operator

$$h:H^n\to H$$

in this closure that is:

• monotone,
• join-continuous,
• fragment-preserving,
• modal and trace-compatible (commutes with $j$ and $G$),

is an admissible behavior.

Definition 5.2 (Fusion-generated operators)

We define ${\text{Op}}_X^{\mathrm{fus}}$ as the smallest set satisfying:

1. ${\text{Op}}_X\subseteq {\text{Op}}_X^{\mathrm{fus}}$;
2. if $f,g\in {\text{Op}}_X^{\mathrm{fus}}$ and $f{\sim }_{X}g$, then we identify them in ${\text{Op}}_X^{\mathrm{fus}}$ (or, equivalently, we treat them as a single operator up to definitional equality);
3. if $h$ is an admissible behavior as above, we may introduce a new symbol $f_h$ with interpretation $[[f_h]]=h$, and include $f_h$ in ${\text{Op}}_X^{\mathrm{fus}}$.

Notice: new operators added by naming are semantic reifications of behaviors that are already modal, trace-compatible, join-continuous, and fragment-preserving. This ensures Modal Adequacy is preserved.

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6. Fusion as a Reflection

The fusion operator we have described is not just some closure: it is a reflection into a full subcategory of “fusion-saturated” structures.

6.1 Fusion Closure Operator ${\mathcal{S}}_{\mathrm{fus}}$

Define

$${\mathcal{S}}_{\mathrm{fus}}:\mathcal{U}\to \mathcal{U}$$

by

$${\mathcal{S}}_{\mathrm{fus}}(X)=(H_X,{\text{Op}}_X^{\mathrm{fus}}),$$

leaving $H_X$ unchanged and saturating ${\text{Op}}_X$ as in Definition 5.2.

Lemma 6.1 (Inflationary and monotone)

${\mathcal{S}}_{\mathrm{fus}}$ is inflationary and monotone:

• $X\le {\mathcal{S}}_{\mathrm{fus}}(X)$ for all $X$;
• if $X\le Y$, then ${\mathcal{S}}_{\mathrm{fus}}(X)\le {\mathcal{S}}_{\mathrm{fus}}(Y)$.

Sketch. Inflationary is by (1) of Definition 5.2. Monotonicity holds because enlarging $H_X$ and ${\text{Op}}_X$ only increases the space of fragments and admissible behaviors, never removing existing ones.

Lemma 6.2 (Idempotence of fusion)

$${\mathcal{S}}_{\mathrm{fus}}({\mathcal{S}}_{\mathrm{fus}}(X))={\mathcal{S}}_{\mathrm{fus}}(X)\text{for all }X\in \mathcal{U}.$$

Proof.
By construction, ${\text{Op}}_X^{\mathrm{fus}}$ is already closed under:

• quotienting by ${\sim }_X$,
• naming admissible behaviors.

If we apply the same construction again to $(H_X,{\text{Op}}_X^{\mathrm{fus}})$, we do not obtain any new identifications (the congruence is already saturated) or new admissible behaviors (all such behaviors have already been named, subject to the same conditions). Thus ${\text{Op}}_X^{\mathrm{fus}}$ is fixed under fusion, and so is $(H_X,{\text{Op}}_X^{\mathrm{fus}})$.

Hence ${\mathcal{S}}_{\mathrm{fus}}$ is a closure operator on the poset $\mathcal{U}$ in the order-theoretic sense: monotone, inflationary, idempotent.

6.2 Fusion-Saturated Structures and Reflection

Let ${\mathcal{U}}_{\mathrm{fus}}$ be the set of fixed points of ${\mathcal{S}}_{\mathrm{fus}}$:

$${\mathcal{U}}_{\mathrm{fus}}:=\{X\in \mathcal{U}:{\mathcal{S}}_{\mathrm{fus}}(X)=X\}.$$

These are the fusion-saturated structures: they contain all identifications and named operators required by fragmentwise semantic behavior.

We regard $\mathcal{U}$ and ${\mathcal{U}}_{\mathrm{fus}}$ as poset-categories (thin categories). There is an inclusion functor

$$i:{\mathcal{U}}_{\mathrm{fus}}\hookrightarrow \mathcal{U}.$$

Lemma 6.3 (Fusion is a reflector)

${\mathcal{S}}_{\mathrm{fus}}$ is left adjoint to $i$. That is, for each $X\in \mathcal{U}$ and each $Y\in {\mathcal{U}}_{\mathrm{fus}}$,

$${\mathcal{S}}_{\mathrm{fus}}(X)\le Y\text{iff}X\le Y.$$

Proof.
In any poset $(P,\le )$, a closure operator $c:P\to P$ has the property that $c(x)\le y$ iff $x\le y$ for all fixed points $y$ of $c$. This is standard: if $x\le y$ and $c(y)=y$, then $c(x)\le c(y)=y$; conversely, if $c(x)\le y$, then $x\le c(x)\le y$.

Applying this to $P=\mathcal{U}$ and $c={\mathcal{S}}_{\mathrm{fus}}$, and noting that ${\mathcal{U}}_{\mathrm{fus}}$ is exactly the set of fixed points, proves the adjunction.

Thus ${\mathcal{S}}_{\mathrm{fus}}$ is the left adjoint $L_{\mathrm{fus}}$ to the inclusion $i$, and ${\mathcal{U}}_{\mathrm{fus}}$ is a reflective subcategory of $\mathcal{U}$.

This is the Fusion Reflection Lemma: fusion is not just closure, but a genuine reflection.

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7. Semantic and Syntactic Closure Operators

We now define the semantic and syntactic closures, respecting finitary dependence.

7.1 Semantic Closure ${\mathcal{S}}_{\mathrm{sem}}$

For $X=(H_X,{\text{Op}}_X)\in \mathcal{U}$, define

$${\mathcal{S}}_{\mathrm{sem}}(X)=(H_X^{\prime },{\text{Op}}_X),$$

where $H_X^{\prime }$ is the smallest subset of $H$ such that:

1. $H_X\subseteq H_X^{\prime }$;
2. (Interpretation closure) If $f\in {\text{Op}}_X$ and $a_i\in H_X^{\prime }$, then $[[f]](a_1,\dots ,a_n)\in H_X^{\prime }$;
3. (Heyting closure) $H_X^{\prime }$ is closed under $\wedge ,\vee ,\Rightarrow ,\perp ,\top$;
4. (Modal–trace closure) $H_X^{\prime }$ is closed under $j$ and $G$;
5. (Fixed-point closure) if $h:H\to H$ is monotone, join-continuous, fragment-preserving, modal, trace-compatible, and lies in the hereditarily extensional closure of $\{[[f]]:f\in {\text{Op}}_X\}$, then any least or greatest fixed point of $h$ that is above some element of $H_X^{\prime }$ is included in $H_X^{\prime }$.

Lemma 7.1

${\mathcal{S}}_{\mathrm{sem}}$ is inflationary and monotone on $\mathcal{U}$.

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7.2 Syntactic Closure with Finitary Dependence ${\mathcal{S}}_{\mathrm{syn}}$

We now make the compactness property precise.

Definition 7.2 (Semantic finitary justification)

For $X=(H_X,{\text{Op}}_X)$, operator $f\in {\mathsf{Op}}^{\mathrm{def}}$, and a fragment $F\subseteq H_X$, we say:

$f$ is semantically justified by $F$ over $X$ if there exists $g\in {\text{Op}}_X$ such that

$$[[f]]{\equiv }_F[[g]]$$

and this equality is hereditarily extensional under:

• Heyting operations,
• $j$, $G$,
• all operators in ${\text{Op}}_X$.

This captures that $f$ behaves like an existing operator on a finite piece of semantics.

Definition 7.3 (Syntactic closure)

For $X=(H_X,{\text{Op}}_X)$, define

$${\mathcal{S}}_{\mathrm{syn}}(X)=(H_X,{\text{Op}}_X^{\prime }),$$

where ${\text{Op}}_X^{\prime }$ is the smallest set such that:

1. ${\text{Op}}_X\subseteq {\text{Op}}_X^{\prime }$;
2. ${\text{Op}}_X^{\prime }$ is closed under λ-definability and composition;
3. if the λ-calculus includes fixed-point constructs whose interpretations are monotone, join-continuous, fragment-preserving, modal, and trace-compatible, these fixed-point operators are included whenever their interpretations arise from ${\text{Op}}_X^{\prime }$;
4. (Finitary justification) if $f\in {\mathsf{Op}}^{\mathrm{def}}$ is semantically justified by some finite fragment $F\subseteq H_X$, then $f\in {\text{Op}}_X^{\prime }$.

Algebraic compactness

By (4), we have:

$${\mathcal{S}}_{\mathrm{syn}}(X)=\bigcup _{F{\subseteq }_{\mathrm{fin}}{H}_X}{\mathcal{S}}_{\mathrm{syn}}(F,{\text{Op}}_X),$$

where syntactic closure over $F$ only uses semantic facts inside the fragment $F$.

Lemma 7.4

${\mathcal{S}}_{\mathrm{syn}}$ is inflationary and monotone.

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7.3 Global Semiotic Closure $\mathcal{S}$ and Fixed Point

Define the global operator

$$\mathcal{S}:={\mathcal{S}}_{\mathrm{fus}}\circ {\mathcal{S}}_{\mathrm{syn}}\circ {\mathcal{S}}_{\mathrm{sem}}:\mathcal{U}\to \mathcal{U}.$$

Lemma 7.5

$\mathcal{S}$ is monotone and inflationary.

Proof. Composition of monotone inflationary maps.

Theorem 7.6 (Existence of least fixed point)

$\mathcal{S}$ has a least fixed point $X^{\star }$ in $\mathcal{U}$, given by

$$X^{\star }=\bigwedge \{X\in \mathcal{U}:\mathcal{S}(X)\le X\}.$$

Proof. $\mathcal{U}$ is a complete lattice; by Knaster–Tarski, any monotone endomap has a nonempty complete lattice of fixed points, with a least element.

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8. The Semiotic Universe and Its Universal Property

8.1 Primitive Data and Base Structure

The primitive semiotic data consist of:

• a complete Heyting algebra $H$, with modality $j$ and trace $G$ as in Section 1;
• a primitive set of syntactic operators ${\text{Op}}^{\mathrm{prim}}\subseteq {\mathsf{Op}}^{\mathrm{def}}$;
• an interpretation of ${\text{Op}}^{\mathrm{prim}}$ satisfying conditions of Section 4 (including fragment preservation).

We embed this into a base partial structure

$$X_0=(H_0,{\text{Op}}_0)$$

where $H_0\subseteq H$ is any chosen generating set for the Heyting–modal–comonadic structure, and ${\text{Op}}_0={\text{Op}}^{\mathrm{prim}}$.

We then apply $\mathcal{S}$ repeatedly.

8.2 Definition of the Semiotic Universe

Definition 8.1 (Semiotic universe)

The semiotic universe is the least fixed point

$${\mathcal{U}}_{\mathrm{semiotic}}:=X^{\star }=(H_{\mathrm{semiotic}},{\text{Op}}_{\mathrm{semiotic}})$$

of $\mathcal{S}$ such that $X_0\le X^{\star }$.

By construction:

• $H_{\mathrm{semiotic}}$ is closed under Heyting operations, $j$, $G$, interpreted operators, and fixed points of admissible semantic operators;
• ${\text{Op}}_{\mathrm{semiotic}}$ is closed under λ-definability, composition, finitary justification, fixed-point constructs, and fusion reflection (i.e. it is fusion-saturated).

Moreover, because fusion is a reflection, ${\mathcal{U}}_{\mathrm{semiotic}}$ is fusion-saturated: ${\mathcal{S}}_{\mathrm{fus}}(X^{\star })=X^{\star }$.

8.3 Semiotic Structures and Morphisms

A semiotic structure over the given primitive data consists of:

• a complete Heyting algebra $K$ with modality $j_K$ and trace $G_K$ satisfying analogues of Axioms 1.2–1.4;
• an interpretation $[[-{]]}_K:{\mathsf{Op}}^{\mathrm{def}}\to \mathrm{Hom}(K^n,K)$ satisfying analogues of Section 4;
• a partial semiotic structure $X_K=(H_K,{\text{Op}}_K)\subseteq K\times {\mathsf{Op}}^{\mathrm{def}}$ that is a fixed point of the induced closure operator ${\mathcal{S}}_K$ and extends the primitive data.

A morphism between semiotic structures $(K_1,X_1)$ and $(K_2,X_2)$ is a pair $(h,k)$ where:

• $h:K_1\to K_2$ is a complete Heyting homomorphism preserving $j$ and $G$:

$$h(j_1(a))=j_2(h(a)),h(G_1(a))=G_2(h(a));$$

• $k:{\text{Op}}_1\to {\text{Op}}_2$ is a homomorphism of syntactic algebras that preserves definitional equality and closure operations;
• $h$ and $k$ are compatible with the interpretation:

$$h([[f{]]}_1(a_1,\dots ,a_n))=[[k(f){]]}_2(h(a_1),\dots ,h(a_n)).$$

Two morphisms $(h,k)$, $(h^{\prime },k^{\prime })$ are viewed as 2-equal if they are fragmentwise extensionally equal in the sense of Section 3.

8.4 Initiality of the Semiotic Universe

Theorem 8.2 (Initiality / Free semiotic universe)

The semiotic universe ${\mathcal{U}}_{\mathrm{semiotic}}$ is initial among all semiotic structures over the primitive data:

For any semiotic structure $(K,X_K)$, there exists a unique (up to fragmentwise extensional equality) morphism

$$(h,k):{\mathcal{U}}_{\mathrm{semiotic}}\to (K,X_K).$$

Sketch of proof.

1. Any semiotic structure $(K,X_K)$ is a fixed point of an induced closure operator ${\mathcal{S}}_K$ extending the same primitive data. Thus it is a pre-fixed point in the corresponding lattice.
2. By minimality of $X^{\star }$, we have $X^{\star }\le X_K$.
3. This inclusion yields natural candidates $h,k$ on the semantic and syntactic components, and these maps preserve all structure because each closure step (semantic, syntactic, fusion) is preserved by any structure satisfying the same axioms.
4. Uniqueness follows because any morphism must agree with the inclusion on the primitive data and respect closure operations; but $X^{\star }$ is obtained from the primitive data by these operations alone.

Because fusion is a reflection (Section 6), the initiality extends to the fusion-saturated structures, making ${\mathcal{U}}_{\mathrm{semiotic}}$ the free modal–comonadic Heyting structure with interpreted syntax.

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9. A 2-Category of Semiotic Universes

Equality of morphisms is fundamentally fragmentwise, so semiotic universes form a natural 2-category:

• 0-cells: semiotic universes compatible with the primitive data;
• 1-cells: morphisms $(h,k)$ preserving all structure;
• 2-cells: fragmentwise extensional equalities between 1-cells (hereditarily extensional under the target structure’s semantics).

The semiotic universe ${\mathcal{U}}_{\mathrm{semiotic}}$ is an initial 0-cell in this 2-category, refining Theorem 8.2.

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10. Discussion

We have now filled in the critical structural gaps:

1. Heyting–comonadic coherence:
   The trace operator $G$ is required to be a complete Heyting algebra endomorphism, not merely a join-continuous comonad. This guarantees that fragments and the modal-temporal subalgebra are stable under implication and meet, and that trace-compatible operators behave coherently with the logic.

2. Fragment preservation by interpreted operators:
   We explicitly require all interpreted operators to be fragment-preserving and assemble them into a hereditarily extensional family. This removes implicit dependence on unproven distribution properties and makes fragmentwise reasoning well-typed and robust.

3. Fusion as reflection:
   Fusion is shown to be an idempotent, monotone, inflationary closure operator on the poset $\mathcal{U}$, thereby inducing a reflective subcategory of fusion-saturated structures. The resulting left adjoint $L_{\mathrm{fus}}={\mathcal{S}}_{\mathrm{fus}}$ underpins the universal property of the semiotic universe.

4. Internal dynamics by monotone endomaps:
   Any additional dynamics (for example, internal agents acting over $\mathcal{U}\times \Sigma$) remain within the same order-theoretic discipline: they are monotone operators compatible with fragments and closure, so they inherit the fixed-point and locality properties already established.

The resulting construction is:

• algebraically and order-theoretically coherent,
• explicitly modal and comonadic,
• compact on the syntactic side,
• and reflective on the fusion side.

This semiotic universe is therefore a solid base for implementations that want to treat recursive, modal, and temporal structure in a way that is jointly controlled by syntax, semantics, and their closure.