Abstract Memetics: Capability-Relative Replication and the Evolution of Semantic Universes

Author: Anonymous

Source: https://abstract-memetics.grok.me/

Part 1: Capability-Relative Memetic Replication

Abstract

We formulate memetic replication as a capability-relative notion of semantic persistence inside an ∞-topos. The central structure is a hierarchy of left-exact localizations

$$L_c:\mathcal{E}\longrightarrow\mathcal{E}_c$$

indexed by cognitive capability levels $c$. A memetic presentation $x\in M$ is perceived by an agent of capability $c$ only through its localized image

$$\eta_c(x)\in L_c M.$$

Two presentations are therefore semantically identical relative to $c$ precisely when their localized images are equivalent. A transmission from an agent $a$ to an agent $b$ counts as replication of a meme $m$ when the transmitted realization and the recipient-localized realization of $m$ are connected by an identity path in $L_{\kappa(b)}M$.

We prove that capability-relative semantic equivalence is an equivalence relation, that equivalence at a finer cognitive resolution implies equivalence at every coarser resolution, that the converse fails exactly when the comparison map between capability levels identifies distinct semantic states, and that replication witnesses compose along chains of transmission. We then define realization spaces, cognitive thresholds, mutation relative to capability, and the internal replication correspondence.

The resulting framework makes precise the proposition that memetic identity is not identity of presentation but equivalence after passage through a capability-dependent semantic modality.

Notation

The symbols of the note, in the order they are used.

• $\mathcal{E}$ — Ambient ∞-topos of states, agents, observations, and semantic structure.
• $c \le d$ — Capability d distinguishes at least the semantic structure visible at c.
• $L_c$ — Accessible left-exact localization: the semantic modality at capability c.
• $\eta_c$ — Unit of the localization, carrying a presentation to its c-semantic state.
• $\rho_{dc}$ — Comparison from the finer modality L_d to the coarser modality L_c.
• $[m]_c$ — The c-semantic realization η_c(m) in L_c M.
• $x \simeq_c y$ — Semantic equivalence: the images of x and y are identical in L_c M.
• $T_{ab}$ — Transmission of a semantic state from agent a into the space of agent b.
• $\operatorname{Rep}_{a\to b}(m)$ — Space of paths witnessing that the transmission replicates m.
• $\operatorname{Real}_c(m)$ — Ways an agent at capability c can internally realize the structure of m.
• $\chi(m)$ — Cognitive threshold: the least capability at which m is realizable.

1 Ambient categorical setting

We work in an ∞-topos $\mathcal{E}$. Objects of $\mathcal{E}$ may be interpreted as spaces of possible states, representations, agents, observations, or semantic structures. Mapping objects and identity types are understood homotopy-coherently.

Let $(\mathcal{C},\le)$ be a small partially ordered set of cognitive capability levels. The intended interpretation is that $c\le d$ when capability $d$ can distinguish at least as much semantic structure as capability $c$. Thus increasing capability corresponds to increasing semantic resolution.

Definition 1.1 Capability modality

A capability modality at level $c\in\mathcal{C}$ is an accessible left-exact localization

$$L_c:\mathcal{E}\longrightarrow\mathcal{E}_c\subseteq\mathcal{E}$$

with unit

$$\eta_c:\operatorname{id}_{\mathcal{E}}\Longrightarrow L_c.$$

Left exactness means that $L_c$ preserves finite limits. Equivalently, $\mathcal{E}_c$ is a reflective sub-∞-category of $\mathcal{E}$, and the reflector preserves finite limits.

Definition 1.2 Capability hierarchy

A family $\{L_c\}_{c\in\mathcal{C}}$ is a capability hierarchy if, whenever $c\le d$, there exists a natural comparison transformation

$$\rho_{dc}:L_d\Longrightarrow L_c$$

such that

$$\rho_{dc}\circ\eta_d\simeq\eta_c$$

and

$$L_c L_d\simeq L_c.$$

For $c\le d\le e$, we require the coherence

$$\rho_{ec}\simeq\rho_{dc}\circ\rho_{ed}.$$

Thus

$$L_e X\longrightarrow L_d X\longrightarrow L_c X$$

successively forgets semantic distinctions.

Example 1.3 Truncation hierarchy

The fundamental example is $\mathcal{C}=\mathbb{N}$ with $L_n=\tau_{\le n}$, where $\tau_{\le n}$ denotes $n$-truncation. For $n\le m$,

$$\tau_{\le n}\tau_{\le m}X\simeq\tau_{\le n}X.$$

Hence

$$\tau_{\le m}X\longrightarrow\tau_{\le n}X$$

forgets homotopical information above degree $n$. This is the canonical model of increasing semantic resolution.

2 Agents and cognitive capability

Definition 2.1 Capability-stratified population

A population of agents is a family of objects $\{A_c\}_{c\in\mathcal{C}}$ of $\mathcal{E}$. Its total object is

$$A:=\coprod_{c\in\mathcal{C}}A_c.$$

For an agent $a\in A_c$, we write $\kappa(a)=c$. The function

$$\kappa:A\longrightarrow\mathcal{C}$$

is called the capability assignment. Strictly internally one may replace the external poset $\mathcal{C}$ by a poset object of $\mathcal{E}$, but the externally indexed version suffices for the present results.

3 Memetic presentations and semantic identity

Let $M\in\mathcal{E}$ be an object whose points represent possible memetic presentations. A point

$$m:1\longrightarrow M$$

may represent, for example, an utterance, diagram, procedure, program, behavioral pattern, mathematical statement, narrative, policy, or other transmissible structure. No assumption is made that $M$ is $0$-truncated. Consequently, semantic identity may itself contain nontrivial higher homotopy.

Definition 3.1 Capability-relative semantic state

For $m\in M$ and capability $c$, define the $c$-semantic realization of $m$ to be

$$[m]_c:=\eta_c(m)\in L_c M.$$

Definition 3.2 Capability-relative semantic equivalence

For $x,y\in M$, define $x\simeq_c y$ to mean that the identity type

$$\operatorname{Id}_{L_c M}\bigl(\eta_c(x),\eta_c(y)\bigr)$$

is inhabited. Equivalently,

$$x\simeq_c y \iff \eta_c(x)\simeq\eta_c(y)\quad\text{in }L_c M.$$

We call this *semantic equivalence at capability $c$*.

Proposition 3.3 Equivalence is an equivalence relation

For every fixed $c$, the relation $\simeq_c$ on points of $M$ is reflexive, symmetric, and transitive.

Proof.

Reflexivity follows from the identity path $\operatorname{id}_{\eta_c(x)}:\eta_c(x)=\eta_c(x)$.

Given $p:\eta_c(x)=\eta_c(y)$, path inversion gives $p^{-1}:\eta_c(y)=\eta_c(x)$, so the relation is symmetric.

Given $p:\eta_c(x)=\eta_c(y)$ and $q:\eta_c(y)=\eta_c(z)$, path composition gives $q\circ p:\eta_c(x)=\eta_c(z)$. Therefore $\simeq_c$ is an equivalence relation.

4 Monotonicity with respect to capability

The basic structural result is that finer semantic equivalence implies coarser semantic equivalence.

Theorem 4.1 Downward persistence of semantic equivalence

Let $c\le d$. If $x\simeq_d y$, then $x\simeq_c y$.

Proof.

Suppose $p:\eta_d(x)=\eta_d(y)$ is a path in $L_d M$. Apply the comparison map

$$\rho_{dc,M}:L_d M\longrightarrow L_c M.$$

Functoriality of identity types produces

$$\rho_{dc,M}(p):\rho_{dc,M}(\eta_d(x))=\rho_{dc,M}(\eta_d(y)).$$

By compatibility of the comparison transformation with the localization units,

$$\rho_{dc,M}\circ\eta_d\simeq\eta_c.$$

Hence $\eta_c(x)=\eta_c(y)$, and therefore $x\simeq_c y$.

Corollary 4.2 Nested semantic equivalence classes

For $c\le d$, every $d$-equivalence class is contained in a $c$-equivalence class. Symbolically,

$$[x]_d\subseteq[x]_c.$$

Thus increasing capability can refine semantic equivalence classes but cannot merge classes that were already distinct at lower capability.

Corollary 4.3 Truncation monotonicity

If $n\le m$ and $\tau_{\le m}(x)\simeq\tau_{\le m}(y)$, then $\tau_{\le n}(x)\simeq\tau_{\le n}(y)$.

Proof.

Take $L_n=\tau_{\le n}$ and $L_m=\tau_{\le m}$ in Theorem 4.1.

5 Failure of the converse

The converse to Theorem 4.1 need not hold. A low-capability observer may identify two structures that remain distinguishable at higher capability.

Definition 5.1 Semantic collapse

For $c\le d$, a pair $x,y\in M$ exhibits *semantic collapse from $d$ to $c$* if

$$x\not\simeq_d y$$

but

$$x\simeq_c y.$$

Theorem 5.2 Characterization of semantic collapse

Let $c\le d$. Semantic collapse occurs in $M$ if and only if the comparison map

$$\rho_{dc,M}:L_d M\longrightarrow L_c M$$

has a homotopy fiber containing at least two inequivalent points belonging to the image of $\eta_d:M\longrightarrow L_d M$.

Proof.

Suppose first that semantic collapse occurs. Then there exist $x,y\in M$ such that $\eta_d(x)\not\simeq\eta_d(y)$ but $\eta_c(x)\simeq\eta_c(y)$. Since

$$\rho_{dc,M}\eta_d(x)\simeq\eta_c(x)$$

and

$$\rho_{dc,M}\eta_d(y)\simeq\eta_c(y),$$

their images under $\rho_{dc,M}$ are equivalent. Thus $\eta_d(x)$ and $\eta_d(y)$ determine inequivalent points in a common homotopy fiber of $\rho_{dc,M}$.

Conversely, suppose some homotopy fiber contains inequivalent points $u\simeq\eta_d(x)$ and $v\simeq\eta_d(y)$ coming from $M$. Since $u$ and $v$ lie in the same homotopy fiber,

$$\rho_{dc,M}(u)\simeq\rho_{dc,M}(v).$$

Hence $\eta_c(x)\simeq\eta_c(y)$, while $\eta_d(x)\not\simeq\eta_d(y)$. Therefore semantic collapse occurs.

Corollary 5.3 Criterion for strict refinement

The capability increase $c<d$ strictly refines the semantics of $M$ precisely when $\rho_{dc,M}$ is non-injective on connected components of the localized image of $M$.

Example 5.4 Homotopical collapse

Suppose $M=S^1$. At capability level $L_0=\tau_{\le 0}$, one has

$$\tau_{\le 0}S^1\simeq\ast,$$

so every point becomes semantically indistinguishable. At capability level $L_1=\tau_{\le 1}$, the fundamental loop structure remains. Thus information encoded in path structure can be invisible to a $0$-truncated semantic observer while remaining meaningful to a $1$-truncated observer.

6 Transmission

Semantic equivalence alone does not specify replication. We must also specify a transmission operation.

Definition 6.1 Transmission map

Let $a\in A_c$ and $b\in A_d$. A transmission channel $a\to b$ is a morphism

$$T_{ab}:L_c M\longrightarrow L_d M.$$

The map $T_{ab}$ describes how a memetic state available to $a$ is transformed into a semantic state accessible to $b$. Transmission need not preserve presentations.

Definition 6.2 Replication witness space

Let $m\in M$. The replication witness space for transmission $a\to b$ is

$$\operatorname{Rep}_{a\to b}(m) := \operatorname{Id}_{L_d M} \bigl( T_{ab}(\eta_c(m)),\, \eta_d(m) \bigr),$$

where $c=\kappa(a)$ and $d=\kappa(b)$.

Definition 6.3 Successful replication

The meme $m$ *replicates successfully from $a$ to $b$* if $\operatorname{Rep}_{a\to b}(m)$ is inhabited. Thus replication means

$$T_{ab}(\eta_c(m))\simeq\eta_d(m).$$

Remark 6.4 Presentation is not semantic identity

Successful replication does not require equality of underlying presentations in $M$. It requires only equality in the recipient's capability-localized semantic space. Thus $m'\neq m$ may nevertheless count as successful replication if $\eta_d(m')\simeq\eta_d(m)$. This formalizes the distinction between presentation identity and semantic identity.

7 Replication under composition

Suppose $a\in A_c$, $b\in A_d$, and $e\in A_f$. Let

$$T_{ab}:L_c M\to L_d M$$

and

$$T_{be}:L_d M\to L_f M.$$

Their composite is

$$T_{ae}:=T_{be}\circ T_{ab}.$$

Theorem 7.1 Composition of replication witnesses

If $\operatorname{Rep}_{a\to b}(m)$ and $\operatorname{Rep}_{b\to e}(m)$ are inhabited, then $\operatorname{Rep}_{a\to e}(m)$ is inhabited.

Proof.

Let $p:T_{ab}(\eta_c(m))=\eta_d(m)$ be a replication witness from $a$ to $b$. Applying $T_{be}$ gives

$$T_{be}(p):T_{be}T_{ab}(\eta_c(m))=T_{be}(\eta_d(m)).$$

Let $q:T_{be}(\eta_d(m))=\eta_f(m)$ be a replication witness from $b$ to $e$. Path composition yields

$$q\circ T_{be}(p):T_{be}T_{ab}(\eta_c(m))=\eta_f(m).$$

But $T_{be}T_{ab}=T_{ae}$. Therefore $T_{ae}(\eta_c(m))=\eta_f(m)$, so $\operatorname{Rep}_{a\to e}(m)$ is inhabited.

Corollary 7.2 Iterated replication

For a chain of agents $a_0\to a_1\to\cdots\to a_n$, if $m$ replicates successfully across every edge, then $m$ replicates successfully under the composite transmission $a_0\to a_n$.

8 Mutation

Replication and mutation are not mutually exclusive notions. Mutation depends on which semantic resolution is being used.

Definition 8.1 Capability-relative mutation

Suppose transmission produces a presentation $m'\in M$. Relative to capability $c$, the transmission is:

• presentation-preserving if $m'\simeq m$ in $M$;

• *semantically faithful at $c$* if $m'\simeq_c m$;

• a *$c$-detectable mutation* if $m'\not\simeq_c m$.

Proposition 8.2 Mutation can disappear under coarse-graining

Let $c\le d$. It is possible that $m'\not\simeq_d m$ while $m'\simeq_c m$. In that case the same transmission is classified as mutation by capability $d$ and faithful replication by capability $c$.

Proof.

This is precisely semantic collapse as defined in Definition 5.1.

Corollary 8.3 Capability-relative fidelity

There is no capability-independent binary distinction between faithful copying and mutation unless the family of modalities is jointly conservative on $M$.

Definition 8.4 Joint conservativity

The capability hierarchy is *jointly conservative on $M$* if

$$\bigl(\forall c\in\mathcal{C},\;\eta_c(x)\simeq\eta_c(y)\bigr) \implies x\simeq y.$$

When joint conservativity fails, two genuinely different ambient structures can be indistinguishable to every capability level represented in the population.

9 Cognitive realization

So far we have described semantic discrimination. We now introduce the ability of an agent actually to instantiate a memetic structure. Let $P_m\in\mathcal{E}$ denote a presentation object for meme $m$. For every capability level $c$, let $C_c\in\mathcal{E}_c$ be a cognitive state object describing realizable structures for agents at capability $c$.

Definition 9.1 Realization space

The realization space of $m$ at capability $c$ is

$$\operatorname{Real}_c(m):=\operatorname{Map}_{\mathcal{E}_c}(L_c P_m,\, C_c).$$

The meme is *realizable at capability $c$* if $\operatorname{Real}_c(m)\neq\varnothing$.

Definition 9.2 Capability-monotone realization system

A realization system is capability-monotone if, for every $c\le d$, there is a map

$$\Phi_{cd,m}:\operatorname{Real}_c(m)\longrightarrow\operatorname{Real}_d(m)$$

natural in $m$. This expresses the assumption that a more capable cognitive architecture can realize every structure realizable by a less capable one.

Theorem 9.3 Upward closure of realizability

In a capability-monotone realization system, $\operatorname{Real}_c(m)\neq\varnothing$ and $c\le d$ imply $\operatorname{Real}_d(m)\neq\varnothing$.

Proof.

Choose $r\in\operatorname{Real}_c(m)$. Applying $\Phi_{cd,m}$ gives

$$\Phi_{cd,m}(r)\in\operatorname{Real}_d(m).$$

Thus the latter realization space is inhabited.

Definition 9.4 Cognitive support

Define the cognitive support of $m$ by

$$\Theta(m):=\{\,c\in\mathcal{C}\mid\operatorname{Real}_c(m)\neq\varnothing\,\}.$$

Corollary 9.5 Upward closure of support

If the realization system is capability-monotone, then $\Theta(m)$ is upward closed:

$$c\in\Theta(m),\quad c\le d \implies d\in\Theta(m).$$

Definition 9.6 Cognitive threshold

If $\Theta(m)$ possesses a least element, define

$$\chi(m):=\min\Theta(m).$$

We call $\chi(m)$ the cognitive threshold of $m$. It is the least capability at which the memetic structure can be internally realized.

10 Comprehension versus replication

Realization and replication are logically distinct.

Definition 10.1 Comprehension condition

An agent $b\in A_d$ can comprehend $m$ if $\operatorname{Real}_d(m)$ is inhabited.

Definition 10.2 Replication condition

A transmission from $a$ to $b$ successfully replicates $m$ if $\operatorname{Rep}_{a\to b}(m)$ is inhabited.

Proposition 10.3 Logical independence

Without additional assumptions,

$$\operatorname{Real}_d(m)\neq\varnothing$$

does not imply

$$\operatorname{Rep}_{a\to b}(m)\neq\varnothing,$$

and

$$\operatorname{Rep}_{a\to b}(m)\neq\varnothing$$

does not imply any richer realization property beyond that explicitly encoded in $L_d M$.

Proof.

The first statement follows because realizability concerns the mapping space $\operatorname{Map}(L_d P_m, C_d)$, whereas replication concerns the identity type $T_{ab}(\eta_c(m))=\eta_d(m)$. No morphism relating these two spaces has been assumed.

For the converse, replication identifies localized semantic states but does not imply that the recipient possesses any additional map $L_d P_m\to C_d$ unless such a connection is separately specified. Thus the two notions are independent in the bare theory.

11 Internal replication correspondence

The local definitions can be assembled into a global object. Assume transmission maps vary coherently with pairs of agents. Define an object

$$\mathcal{R}\longrightarrow A\times A\times M$$

whose homotopy fiber over $(a,b,m)$ is $\operatorname{Rep}_{a\to b}(m)$.

Definition 11.1 Memetic replication correspondence

The span

$$A\times M \xleftarrow{s} \mathcal{R} \xrightarrow{t} A\times M$$

is the memetic replication correspondence. The source map records $(a,m)$, while the target map records the corresponding recipient state $(b,m)$. More general versions may allow the target presentation to be $m'$, producing $(a,m)\rightsquigarrow(b,m')$.

Definition 11.2 Generalized replication correspondence

Define

$$\widetilde{\mathcal{R}}\longrightarrow A\times A\times M\times M$$

with fiber

$$\widetilde{\mathcal{R}}_{a,b,m,m'} := \operatorname{Id}_{L_{\kappa(b)}M} \bigl( T_{ab}(\eta_{\kappa(a)}m),\, \eta_{\kappa(b)}m' \bigr).$$

Then $(a,m)\rightsquigarrow(b,m')$ is a valid transmission exactly when this identity type is inhabited. Faithful replication of $m$ corresponds to the diagonal case $m'=m$. Capability-relative mutation corresponds to $m'\neq m$ while $m'\simeq_{\kappa(b)} m$.

12 Replication as a homotopy-coherent dynamical system

A sequence

$$(a_0,m_0)\rightsquigarrow(a_1,m_1)\rightsquigarrow\cdots\rightsquigarrow(a_n,m_n)$$

defines a path in the correspondence-generated dynamics. At each stage we may have $m_i\not\simeq m_{i+1}$ in the ambient object $M$, even though $m_i\simeq_{\kappa(a_{i+1})}m_{i+1}$. Thus memetic persistence is naturally represented not by a constant point of $M$, but by a coherently related sequence of points whose images agree after appropriate localization.

Definition 12.1 Capability-relative memetic lineage

A memetic lineage is a sequence $(m_i,a_i)_{i\in I}$ together with transmission witnesses

$$p_i: T_{a_i a_{i+1}} \bigl(\eta_{\kappa(a_i)}m_i\bigr) = \eta_{\kappa(a_{i+1})}m_{i+1}.$$

Definition 12.2 Stable memetic lineage

Such a lineage is semantically stable relative to the recipients if

$$m_{i+1}\simeq_{\kappa(a_{i+1})}m_i$$

for every $i$. It need not be stable in the ambient space $M$.

13 The central representation theorem

We may now state the basic conceptual result of the framework.

Theorem 13.1 Capability-relative criterion for memetic identity

Let $m,m'\in M$. For an agent $a\in A_c$, the two presentations $m,m'$ represent the same memetic state accessible to $a$ if and only if

$$\eta_c(m)\simeq\eta_c(m')$$

in $L_c M$. Equivalently, $m\simeq_c m'$.

Proof.

By construction, all semantic states available at capability $c$ factor through the localization $\eta_c:M\to L_c M$. Hence an agent at capability $c$ has access only to the images $\eta_c(m)$ and $\eta_c(m')$. If these images are equivalent, they determine the same state of $L_c M$ up to its intrinsic homotopy notion of identity.

Conversely, if they are inequivalent in $L_c M$, then the localized semantic system contains a distinction between them. Therefore semantic identity relative to capability $c$ is exactly equality in $L_c M$.

Theorem 13.2 Replication preserves semantic identity

A transmission $T_{ab}:L_{\kappa(a)}M\to L_{\kappa(b)}M$ successfully replicates $m$ if and only if

$$T_{ab}\bigl([m]_{\kappa(a)}\bigr)\simeq[m]_{\kappa(b)}.$$

Proof.

By Definition 6.2,

$$\operatorname{Rep}_{a\to b}(m) = \operatorname{Id}_{L_{\kappa(b)}M} \bigl( T_{ab}([m]_{\kappa(a)}),\, [m]_{\kappa(b)} \bigr).$$

This space is inhabited precisely when the two displayed points are equivalent.

14 Apparent fidelity and cognitive coarseness

The theory has an immediate consequence relevant to populations with heterogeneous cognitive capabilities.

Theorem 14.1 Coarse observers weakly increase apparent fidelity

Let $c\le d$. If a transmission is semantically faithful at capability $d$, then it is semantically faithful at capability $c$. The converse need not hold.

Proof.

Faithfulness at $d$ means $m'\simeq_d m$. By Theorem 4.1, $m'\simeq_c m$. The converse fails whenever semantic collapse occurs between $d$ and $c$, by Theorem 5.2.

Corollary 14.2 Measured fidelity can rise as resolution falls

A decrease in cognitive resolution can increase the measured rate of faithful replication without increasing presentation-level fidelity. This follows because the equivalence classes $[-]_c$ become coarser as capability decreases.

Remark 14.3 Sloganization and semantic loss

This provides a formal mechanism for phenomena in which many distinct presentations are classified as instances of one simple semantic pattern. Examples may include simplification, sloganization, compression, ritualized repetition, or semantic loss under repeated transmission. The mathematical claim does not depend on any particular empirical theory of human cognition; it follows from the localization structure once such a structure is adopted as a model.

15 Memetic complexity

Capability thresholds induce a natural notion of semantic complexity.

Definition 15.1 Capability complexity

Suppose $\mathcal{C}$ is totally ordered and every realizable meme possesses a cognitive threshold. Define

$$\operatorname{Comp}(m):=\chi(m).$$

Thus $\operatorname{Comp}(m)$ is the least capability required to realize $m$.

Proposition 15.2 Complexity monotonicity under realization

If $\operatorname{Comp}(m)=c$, then every $d\ge c$ can realize $m$, assuming capability-monotone realization.

Proof.

Immediate from Theorem 9.3.

16 Semantic compression

The localizations themselves can be interpreted as semantic compression operators. For $c\le d$, the comparison

$$\rho_{dc,M}:L_d M\to L_c M$$

forgets distinctions inaccessible at level $c$.

Definition 16.1 Capability-relative compression

The map $\rho_{dc,M}$ is the *semantic compression map from $d$ to $c$*. Its homotopy fibers measure how many finer semantic states collapse into one coarser state.

Proposition 16.2 Fibers detect collapsed distinctions

If a homotopy fiber of $\rho_{dc,M}$ contains multiple connected components, then capability $c$ identifies semantic states distinguishable at capability $d$.

Proof.

This is the fiberwise form of Theorem 5.2.

17 Information loss as failure of conservativity

A transmission may destroy distinctions.

Definition 17.1 Semantically conservative transmission

A transmission $T:L_c M\to L_d M$ is semantically conservative if it reflects equivalences:

$$T(x)\simeq T(y)\implies x\simeq y.$$

Definition 17.2 Semantic information loss

Transmission exhibits semantic information loss if there exist

$$x\not\simeq y$$

such that

$$T(x)\simeq T(y).$$

Proposition 17.3 Conservativity is absence of loss

A transmission has no semantic information loss if and only if it is conservative on the relevant full sub-∞-groupoid of $L_c M$.

Proof.

This is the definition of conservativity specialized to the semantic states under consideration.

18 Replication, compression, and selection

The foregoing structures distinguish three independent components.

They correspond respectively to the realization space, the existence and properties of the transmission maps, and the identity type witnessing replication. Any quantitative evolutionary or probabilistic theory built on top of the present framework should distinguish at least these three factors. One may schematically write

$$\begin{aligned} &\text{effective replication rate} \\ &\quad\sim \text{exposure} \times \text{realizability} \times \text{transmission probability} \times \text{capability-relative fidelity}, \end{aligned}$$

but such a numerical formula requires additional probabilistic structure not assumed here.

19 Formal specification

The complete deterministic specification may be summarized by the tuple

$$\begin{aligned} \mathfrak{M} &= \bigl( \mathcal{E},\; \mathcal{C},\; \{L_c\}_{c\in\mathcal{C}},\; \{A_c\}_{c\in\mathcal{C}}, \\ &\qquad M,\; \{C_c\}_{c\in\mathcal{C}},\; \{T_{ab}\} \bigr). \end{aligned}$$

• $\mathcal{E}$: Ambient ∞-topos.
• $\mathcal{C}$: Partially ordered capability levels.
• $L_c$: Capability-dependent semantic modality.
• $A_c$: Agents of capability c.
• $M$: Space of memetic presentations.
• $C_c$: Cognitive realization object at level c.
• $T_{ab}$: Transmission morphisms.

The fundamental identity criterion is

$$m\simeq_c m' \iff \operatorname{Id}_{L_c M} \bigl(\eta_c(m),\,\eta_c(m')\bigr) \text{ is inhabited.}$$

The fundamental replication criterion is

$$\operatorname{Rep}_{a\to b}(m) = \operatorname{Id}_{L_{\kappa(b)}M} \bigl( T_{ab}(\eta_{\kappa(a)}m),\, \eta_{\kappa(b)}m \bigr).$$

20 Main structural consequences

The preceding results may be summarized as follows.

• Theorem A — Equivalence. For each capability $c$, the relation $\simeq_c$ is an equivalence relation.

• Theorem B — Capability monotonicity. If $c\le d$, then $x\simeq_d y$ implies $x\simeq_c y$.

• Theorem C — Strict semantic refinement. The converse fails precisely when $L_d M\to L_c M$ collapses distinct semantic states.

• Theorem D — Composition. Replication witnesses compose along transmission chains.

• Theorem E — Capability-relative mutation. A transformation may be mutation at capability $d$ and faithful replication at capability $c<d$.

• Theorem F — Cognitive threshold. Under capability-monotone realizability, the set of capabilities able to realize a meme is upward closed.

21 Interpretation

The principal conceptual claim can now be stated precisely.

A meme is not intrinsically identified with a unique physical, syntactic, or representational presentation. Instead, relative to cognitive capability $c$, its operative identity is its image

$$[m]_c=\eta_c(m)\in L_c M.$$

Hence memetic identity is indexed by capability:

$$m\longmapsto\bigl\{[m]_c\bigr\}_{c\in\mathcal{C}}.$$

Replication likewise cannot be defined solely by duplication of presentations. It is instead preservation of localized semantic identity:

$$T_{ab}\bigl([m]_{\kappa(a)}\bigr)\simeq[m]_{\kappa(b)}.$$

Memetic replication is capability-relative semantic invariance under transmission.

A lower-capability system corresponds to a coarser localization and therefore possesses larger semantic equivalence classes. A higher-capability system corresponds to a finer localization and may distinguish structures previously collapsed together. Thus $c\le d$ induces the refinement

$$[m]_d\mapsto[m]_c,$$

and consequently

$$\begin{gathered} m\simeq_d m' \implies m\simeq_c m', \\[0.65em] m\simeq_c m' \;\text{does not imply}\; m\simeq_d m'. \end{gathered}$$

This provides the formal sense in which memetic replication is relative to cognitive capability.

22 Conclusion

The ∞-topos formulation replaces a naive conception of a meme as an invariant token by a hierarchy of capability-relative semantic equivalence classes.

Given an ambient ∞-topos $\mathcal{E}$, a hierarchy of left-exact localizations $L_c:\mathcal{E}\to\mathcal{E}_c$, and a memetic state object $M$, the identity of a meme for an observer of capability $c$ is represented by its localized state $\eta_c(m)\in L_c M$.

Transmission from $a$ to $b$ constitutes replication exactly when

$$T_{ab}\bigl(\eta_{\kappa(a)}m\bigr)\simeq\eta_{\kappa(b)}m.$$

This produces a homotopy-coherent notion of replication in which copying, mutation, semantic compression, comprehension, and capability are formally separated.

Identity of memes is not identity of presentations.

**

Identity is equivalence in the semantic localization determined by cognitive capability.

Consequently, memetic replication is not an absolute relation on presentations. It is a capability-indexed relation internal to the appropriate localized ∞-topos.

Part 2: Operadic Memetic Replication

Abstract

We extend the theory of observer-relative memetic replication from unary transmission operators to genuinely multi-input recombination. Let

$$\mathcal{H}=\operatorname{Sh}_{\infty}(\mathcal{C},J)$$

be the ∞-topos associated to a cognitive ∞-site. Rather than assuming that a memetic state evolves only by an endomorphism

$$R:M\to M,$$

we equip the memetic field $M\in\mathcal{H}$ with an algebra structure over an ∞-operad $\mathcal{O}$. An $n$-ary operation $o\in\mathcal{O}(n)$ then induces a recombination map

$$\mu_o^n:M^n\longrightarrow M.$$

This permits memes to be copied, composed, abstracted, translated, fused, specialized, compressed, or otherwise transformed by operations of arbitrary finite arity.

Replication histories are consequently no longer linear chains but rooted $\mathcal{O}$-decorated trees. Operadic composition assigns every such tree a coherent composite operation, and the algebra structure evaluates the tree to a memetic state. We prove that semantic equivalence is preserved by all operadic operations, that tree evaluation is invariant under operadic reassociation, that restriction between cognitive architectures commutes with recombination, and that spaces of recombination witnesses satisfy descent.

The free $\mathcal{O}$-algebra provides a canonical space of formal memetic traces. Its evaluation map

$$\operatorname{ev}_M:F_{\mathcal{O}}(UM)\longrightarrow M$$

identifies syntactically or causally different histories that realize the same semantic state. This yields a precise distinction between operational history and semantic identity.

The unary theory of replication, developed in the preceding note, is recovered as a special case.

Notation

The symbols of this note, in the order they become active.

• $(\mathcal{C},J)$ — Cognitive ∞-site. Architectures are objects of C; J is the Grothendieck topology.
• $\mathcal{H}$ — The ∞-topos of sheaves Sh_∞(C, J), with cartesian product.
• $M(c)$ — Memetic states distinguishable at architecture c.
• $u^*$ — Restriction along a morphism u: c → d, from states at d to states at c.
• $\mathcal{O}(n)$ — The space of n-ary memetic operations.
• $\mu_n$ — The algebra action O(n) × M^n → M.
• $x\simeq_c y$ — Semantic identity: a path in M(c).
• $\operatorname{Rec}_c$ — Witness that an operation on given inputs yields a stated output.
• $\operatorname{Rep}^{(j)}$ — Witness that the j-th parent is reproduced.
• $F_{\mathcal{O}}$ — Free O-algebra: formal finite operadic expressions.
• $\epsilon_M$ — Trace evaluation, from formal histories to semantic states.

1 Ambient cognitive semantics

Let $(\mathcal{C},J)$ be a cognitive ∞-site and put

$$\mathcal{H}:=\operatorname{Sh}_{\infty}(\mathcal{C},J).$$

We use the cartesian symmetric monoidal structure

$$(\mathcal{H},\times,1).$$

Let $M\in\mathcal{H}$ be a memetic field. For every cognitive architecture $c\in\mathcal{C}$, the space $M(c)$ is interpreted as the space of memetic states distinguishable at $c$.

For every architecture morphism $u:c\to d$ there is a restriction map

$$u^*:M(d)\longrightarrow M(c).$$

2 Internal ∞-operads of memetic transformations

We require a structure encoding operations of arbitrary finite arity.

Definition 2.1 Internal memetic operad

An internal memetic ∞-operad $\mathcal{O}$ in $\mathcal{H}$ consists, schematically, of objects

$$\mathcal{O}(n)\in\mathcal{H},\qquad n\ge 0,$$

equipped with symmetric-group actions, units, and homotopy-coherent substitution maps

$$\gamma: \mathcal{O}(k)\times\mathcal{O}(n_1)\times\cdots\times\mathcal{O}(n_k) \longrightarrow \mathcal{O}(n_1+\cdots+n_k)$$

satisfying the usual operadic associativity, equivariance, and unit axioms up to coherent higher homotopy.

A section $o\in\mathcal{O}(n)(c)$ is interpreted as an $n$-ary transformation available at architecture $c$.

Examples include:

• copy

• concatenate

• summarize

• translate

• generalize

• specialize

• compare

• combine

• infer

The mathematics does not privilege any particular interpretation.

3 Memetic algebras

Definition 3.1 O-algebra

An $\mathcal{O}$-memetic algebra is a memetic field $M\in\mathcal{H}$ together with morphisms

$$\mu_n:\mathcal{O}(n)\times M^n\longrightarrow M$$

compatible with the operadic composition, symmetry, and unit structure.

For $o\in\mathcal{O}(n)(c)$ and $x_1,\ldots,x_n\in M(c)$, we write

$$\mu_c(o;x_1,\ldots,x_n)\in M(c).$$

The basic dynamics is therefore

$$(x_1,\ldots,x_n)\overset{o}{\longmapsto}\mu(o;x_1,\ldots,x_n).$$

Unary transmission is merely the case $n=1$. Binary recombination is

$$\mu:\mathcal{O}(2)\times M\times M\to M.$$

4 Semantic identity

For $x,y\in M(c)$, define $x\simeq_c y$ when

$$\operatorname{Id}_{M(c)}(x,y)$$

is inhabited. As before, semantic identity is therefore architecture-relative and homotopical.

5 Operadic operations respect semantic identity

The first structural theorem follows simply from functoriality.

Theorem 5.1 Homotopy invariance of recombination

Let $o,o'\in\mathcal{O}(n)(c)$ and $x_i,x_i'\in M(c)$. Suppose $o\simeq o'$ and, for every $i$, $x_i\simeq_c x_i'$. Then

$$\mu_c(o;x_1,\ldots,x_n) \simeq_c \mu_c(o';x_1',\ldots,x_n').$$

Proof.

The algebra structure is a map of spaces $\mu_{n,c}:\mathcal{O}(n)(c)\times M(c)^n\to M(c)$. The assumed equivalences give a path

$$(o,x_1,\ldots,x_n)\simeq(o',x_1',\ldots,x_n')$$

in its domain. Applying $\mu_{n,c}$ yields a path

$$\mu_c(o;x_1,\ldots,x_n) \simeq \mu_c(o';x_1',\ldots,x_n')$$

in $M(c)$.

Corollary 5.2 Semantic congruence

Capability-relative semantic equivalence is a congruence with respect to every available memetic operation. Thus

$$x_i\simeq_c x_i'\quad\forall i$$

implies

$$o(x_1,\ldots,x_n)\simeq_c o(x_1',\ldots,x_n').$$

This is the operadic analogue of substitutivity of equality.

6 Recombination witnesses

Definition 6.1 Recombination witness space

For $o\in\mathcal{O}(n)(c)$ and $x_1,\ldots,x_n,y\in M(c)$, define

$$\operatorname{Rec}_c(o;x_1,\ldots,x_n;y)$$

by

$\operatorname{Rec}_c(o;x_1,\ldots,x_n;y) := \operatorname{Id}_{M(c)} \bigl(\mu_c(o;x_1,\ldots,x_n),\, y\bigr).$

The recombination succeeds with output $y$ when this space is inhabited. The definition separates inputs, operation, output, and the witness of semantic agreement.

7 Replication as a special kind of recombination

A recombination need not preserve any input. Replication is obtained by requiring the output to remain in the semantic class of some designated parent or target.

Definition 7.1 Parent-relative replication

Let $1\le j\le n$. Define $\operatorname{Rep}^{(j)}_c(o;x_1,\ldots,x_n)$ by

$$\operatorname{Id}_{M(c)} \bigl(\mu_c(o;x_1,\ldots,x_n),\, x_j\bigr).$$

If this space is inhabited, the $j$-th input has reproduced its semantic type through the operation $o$.

Definition 7.2 Target-relative replication

For a distinguished semantic target $m\in M(c)$, define

$$\operatorname{Id}_{M(c)}\bigl(\mu_c(o;\vec x),\, m\bigr).$$

This is useful when the output need not be identical to any particular input.

8 Unary replication is recovered

Let $r\in\mathcal{O}(1)$. The corresponding operation is

$$R=\mu_r:M\to M.$$

Then parent-relative replication is the identity type

$$\operatorname{Id}_{M(c)}(R_c(x),x).$$

Thus the previous fixed-point theory $\operatorname{Eq}(R,\operatorname{id}_M)$ is exactly the unary subtheory of the operadic construction.

9 Operadic substitution

Suppose $o\in\mathcal{O}(k)$ and $o_i\in\mathcal{O}(n_i)$ for $1\le i\le k$. Their operadic composite is

$$o\circ(o_1,\ldots,o_k)\in\mathcal{O}(n_1+\cdots+n_k).$$

The algebra axioms provide a coherent equivalence

$$\begin{aligned} &\mu\bigl(o;\,\mu(o_1;\vec x_1),\ldots,\mu(o_k;\vec x_k)\bigr) \\ &\qquad\simeq \mu\bigl(o\circ(o_1,\ldots,o_k);\,\vec x_1,\ldots,\vec x_k\bigr). \end{aligned}$$

This equation is the formal mechanism by which several generations of recombination collapse to one composite operation.

10 Tree-structured memetic histories

Operadic composition is naturally encoded by rooted trees.

Definition 10.1 O-decorated memetic tree

An $\mathcal{O}$-decorated memetic tree $T$ consists of a finite rooted tree; an operation $o_v\in\mathcal{O}(n_v)$ assigned to every internal vertex $v$ of arity $n_v$; and memetic states assigned to the leaves. The root represents the final output.

Definition 10.2 Evaluation of a memetic tree

The evaluation $\operatorname{ev}_T$ is obtained recursively by applying the operation at every vertex to the evaluations of its incoming branches. Thus

$$\operatorname{ev}_T(\vec x)\in M.$$

11 Tree contraction theorem

Theorem 11.1 Operadic contraction of replication histories

For every finite $\mathcal{O}$-decorated rooted tree $T$, there exists a composite operation $o_T\in\mathcal{O}(N)$, where $N$ is the number of leaves, such that

$$\operatorname{ev}_T(x_1,\ldots,x_N) \simeq \mu(o_T;x_1,\ldots,x_N).$$

The composite $o_T$ is well defined up to the coherent equivalence supplied by the ∞-operad structure.

Proof.

Proceed by induction on the number of internal vertices.

If $T$ has one internal vertex, the result is immediate.

Suppose the root is decorated by $o\in\mathcal{O}(k)$ and its $k$ input subtrees are $T_1,\ldots,T_k$. By induction, each subtree determines $o_{T_i}\in\mathcal{O}(n_i)$ such that $\operatorname{ev}_{T_i}\simeq\mu(o_{T_i};-)$. Therefore

$$\operatorname{ev}_T \simeq \mu\bigl(o;\,\mu(o_{T_1};-),\ldots,\mu(o_{T_k};-)\bigr).$$

By the algebra compatibility with operadic substitution,

$$\operatorname{ev}_T \simeq \mu\bigl(o\circ(o_{T_1},\ldots,o_{T_k});-\bigr).$$

Define $o_T:=o\circ(o_{T_1},\ldots,o_{T_k})$. Higher associativity coherence of the ∞-operad guarantees independence, up to canonical coherent equivalence, from choices of recursive decomposition.

12 Lineages become trees rather than chains

In the unary theory a history had the form

$$x_0\to x_1\to x_2\to\cdots.$$

In the operadic theory one instead has branching and merging:

$$\begin{array}{ccccc} x_1 && x_2 && x_3 \\ & \searrow & \downarrow & \swarrow & \\ && y && \\ & \swarrow && \searrow & \\ z_1 &&&& z_2 \\ & \searrow && \swarrow & \\ && w && \end{array}$$

The appropriate causal object is therefore a rooted tree or, after allowing sharing and feedback, a directed acyclic graph or higher categorical network.

13 Semantic identity of histories

Two distinct causal histories can produce the same semantic result.

Definition 13.1 Semantic equivalence of histories

Let $T$ and $T'$ be two operadic histories with outputs $\operatorname{ev}_T(\vec x)$ and $\operatorname{ev}_{T'}(\vec y)$ at architecture $c$. Define $T\simeq_c T'$ if

$$\operatorname{ev}_T(\vec x)\simeq_c\operatorname{ev}_{T'}(\vec y).$$

Thus history identity is strictly finer than semantic identity.

14 The free operadic algebra

Assume the forgetful functor

$$U:\operatorname{Alg}_{\mathcal{O}}(\mathcal{H})\longrightarrow\mathcal{H}$$

admits a left adjoint

$$F_{\mathcal{O}}:\mathcal{H}\longrightarrow\operatorname{Alg}_{\mathcal{O}}(\mathcal{H}).$$

This holds in the standard presentable setting for suitably small operads.

The object $F_{\mathcal{O}}(X)$ is the free $\mathcal{O}$-algebra generated by $X$. Its elements are formal finite operadic expressions built from points of $X$. Schematically,

$$x,\qquad o(x_1,\ldots,x_n),\qquad o\bigl(o_1(\cdots),\ldots,o_k(\cdots)\bigr),$$

and so forth.

15 Formal memetic traces

Definition 15.1 Trace object

For an $\mathcal{O}$-algebra $M$, define its formal trace algebra

$$\operatorname{Tr}_{\mathcal{O}}(M):=F_{\mathcal{O}}(UM).$$

A point of this object represents a finite formal history of operadic transformations whose leaves are memetic states. The counit of the free–forgetful adjunction gives an $\mathcal{O}$-algebra morphism

$$\epsilon_M:F_{\mathcal{O}}(UM)\longrightarrow M.$$

We call this the trace evaluation map:

$$\operatorname{ev}_M:\operatorname{Tr}_{\mathcal{O}}(M)\longrightarrow M.$$

16 Trace-to-semantic-class map

Definition 16.1 Semantic equivalence of traces

For traces $t,t'\in\operatorname{Tr}_{\mathcal{O}}(M)(c)$, define $t\equiv_c t'$ if

$$\operatorname{ev}_M(t)\simeq_c\operatorname{ev}_M(t').$$

Equivalently,

$$\operatorname{Id}_{M(c)} \bigl(\operatorname{ev}_M(t),\,\operatorname{ev}_M(t')\bigr)$$

is inhabited. This gives the fundamental map

formal transformation history maps to a semantic operation class.

Many distinct traces can map to the same semantic class. Hence

$$t\neq t'$$

does not imply

$$\operatorname{ev}(t)\not\simeq\operatorname{ev}(t').$$

17 Universal property of the trace algebra

Theorem 17.1 Universal evaluation theorem

Let $M$ be an $\mathcal{O}$-algebra. Every map of underlying objects $f:X\to UM$ extends, uniquely up to contractible choice, to an $\mathcal{O}$-algebra morphism

$$\bar f:F_{\mathcal{O}}(X)\to M.$$

Proof.

This is the defining adjunction $F_{\mathcal{O}}\dashv U$. Explicitly,

$$\operatorname{Map}_{\operatorname{Alg}_{\mathcal{O}}} (F_{\mathcal{O}}(X),M) \simeq \operatorname{Map}_{\mathcal{H}}(X,UM).$$

The homotopy fiber over $f$ is therefore contractible.

Corollary 17.2 Canonical trace evaluation

Taking $X=UM$ and $f=\operatorname{id}_{UM}$ produces the canonical trace evaluation map

$$\epsilon_M:F_{\mathcal{O}}(UM)\to M.$$

18 Operational nonuniqueness

The trace formalism gives an immediate nonuniqueness theorem.

Theorem 18.1 Semantic equality does not imply historical equality

Suppose the evaluation map $\epsilon_M:F_{\mathcal{O}}(UM)\to M$ has a homotopy fiber with more than one connected component. Then there exist formally inequivalent histories $t\not\simeq t'$ such that $\epsilon_M(t)\simeq\epsilon_M(t')$.

Proof.

Choose $t,t'$ in distinct connected components of one homotopy fiber of $\epsilon_M$. By definition of the fiber, $\epsilon_M(t)\simeq\epsilon_M(t')$. But $t\not\simeq t'$ in the free algebra.

Thus uniqueness of semantic output never implies uniqueness of causal or computational realization unless the evaluation map is appropriately conservative. See also Corollary 17.2.

19 Architecture restriction commutes with recombination

Because $\mu_n:\mathcal{O}(n)\times M^n\to M$ is a morphism of sheaves, it is natural in the architecture.

Theorem 19.1 Recombination commutes with cognitive restriction

Let $u:c\to d$. Then

$$u^*\mu_d(o;x_1,\ldots,x_n) \simeq \mu_c(u^*o;\,u^*x_1,\ldots,u^*x_n).$$

Proof.

Naturality of the sheaf morphism $\mu_n:\mathcal{O}(n)\times M^n\to M$ gives the commutative square

$$\begin{array}{ccc} \mathcal{O}(n)(d)\times M(d)^n & \xrightarrow{\mu_d} & M(d) \\ \downarrow && \downarrow \\ \mathcal{O}(n)(c)\times M(c)^n & \xrightarrow{\mu_c} & M(c). \end{array}$$

The stated equivalence follows.

20 Observer-relative recombination

The preceding theorem implies that performing a transformation and then restricting is equivalent to restricting the inputs and the operation, and then performing the transformation. Thus the operadic dynamics is compatible with observer relativity.

Corollary 20.1 Restriction of a successful recombination

If $\mu_d(o;\vec x)\simeq_d y$, then

$$\mu_c(u^*o;u^*\vec x)\simeq_c u^*y.$$

Proof.

Apply $u^*$ to the original equivalence and use Theorem 19.1.

21 Apparent fidelity under semantic collapse

Suppose $u:c\to d$ is nonconservative. It may happen that $\mu_d(o;\vec x)\not\simeq_d y$ while $u^*\mu_d(o;\vec x)\simeq_c u^*y$. Using Theorem 19.1,