A formal note

Capability-relative memetic replication in an ∞-topos

Memetic identity is not identity of presentation. It is equivalence after passage through a capability-dependent semantic modality.

∞-topos / left-exact localization / semantic identity / transmission

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} : E ⟶ 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

η_{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 $L_{κ (b)} M$ .

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.

$E$: Ambient ∞-topos of states, agents, observations, and semantic structure.
$c \leq d$: Capability d distinguishes at least the semantic structure visible at c.
$L_{c}$: Accessible left-exact localization: the semantic modality at capability c.
$η_{c}$: Unit of the localization, carrying a presentation to its c-semantic state.
$ρ_{d c}$: 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.
$x ≃_{c} y$: 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.
$Rep_{a \to b} (m)$: Space of paths witnessing that the transmission replicates m.
$Real_{c} (m)$: Ways an agent at capability c can internally realize the structure of m.
$χ (m)$: Cognitive threshold: the least capability at which m is realizable.

1Ambient categorical setting

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

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

Definition1.1Capability modality

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

L_{c} : E ⟶ E_{c} \subseteq E

with unit

η_{c} : id_{E} ⟹ L_{c} .

Left exactness means that $L_{c}$ preserves finite limits. Equivalently, $E_{c}$ is a reflective sub-∞-category of $E$ , and the reflector preserves finite limits.

Definition1.2Capability hierarchy

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

ρ_{d c} : L_{d} ⟹ L_{c}

such that

ρ_{d c} \circ η_{d} ≃ η_{c}

and

L_{c} L_{d} ≃ L_{c} .

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

ρ_{ec} ≃ ρ_{d c} \circ ρ_{e d} .

Thus

L_{e} X ⟶ L_{d} X ⟶ L_{c} X

successively forgets semantic distinctions.

Example1.3Truncation hierarchy

The fundamental example is $C = N$ with $L_{n} = τ_{\leq n}$ , where $τ_{\leq n}$ denotes $n$ -truncation. For $n \leq m$ ,

τ_{\leq n} τ_{\leq m} X ≃ τ_{\leq n} X .

Hence

τ_{\leq m} X ⟶ τ_{\leq n} X

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

2Agents and cognitive capability

Definition2.1Capability-stratified population

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

A := c \in C ∐ A_{c} .

For an agent $a \in A_{c}$ , we write $κ (a) = c$ . The function

κ : A ⟶ C

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

3Memetic presentations and semantic identity

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

m : 1 ⟶ 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.

Definition3.1Capability-relative semantic state

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

[m]_{c} := η_{c} (m) \in L_{c} M .

Definition3.2Capability-relative semantic equivalence

For $x, y \in M$ , define $x ≃_{c} y$ to mean that the identity type

Id_{L_{c} M} (η_{c} (x), η_{c} (y))

is inhabited. Equivalently,

x ≃_{c} y ⟺ η_{c} (x) ≃ η_{c} (y) in L_{c} M .

We call this semantic equivalence at capability $c$ .

Proposition3.3Equivalence is an equivalence relation

For every fixed $c$ , the relation $≃_{c}$ on points of $M$ is reflexive, symmetric, and transitive.

Proof.

Reflexivity follows from the identity path $id_{η_{c} (x)} : η_{c} (x) = η_{c} (x)$ .

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

Given $p : η_{c} (x) = η_{c} (y)$ and $q : η_{c} (y) = η_{c} (z)$ , path composition gives $q \circ p : η_{c} (x) = η_{c} (z)$ . Therefore $≃_{c}$ is an equivalence relation.

4Monotonicity with respect to capability

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

Theorem4.1Downward persistence of semantic equivalence

Let $c \leq d$ . If $x ≃_{d} y$ , then $x ≃_{c} y$ .

Proof.

Suppose $p : η_{d} (x) = η_{d} (y)$ is a path in $L_{d} M$ . Apply the comparison map

ρ_{d c, M} : L_{d} M ⟶ L_{c} M .

Functoriality of identity types produces

ρ_{d c, M} (p) : ρ_{d c, M} (η_{d} (x)) = ρ_{d c, M} (η_{d} (y)) .

By compatibility of the comparison transformation with the localization units,

ρ_{d c, M} \circ η_{d} ≃ η_{c} .

Hence $η_{c} (x) = η_{c} (y)$ , and therefore $x ≃_{c} y$ .

Corollary4.2Nested semantic equivalence classes

For $c \leq d$ , 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.

Corollary4.3Truncation monotonicity

If $n \leq m$ and $τ_{\leq m} (x) ≃ τ_{\leq m} (y)$ , then $τ_{\leq n} (x) ≃ τ_{\leq n} (y)$ .

Proof.

Take $L_{n} = τ_{\leq n}$ and $L_{m} = τ_{\leq m}$ in Theorem 4.1.

5Failure 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.

Definition5.1Semantic collapse

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

x \neq ≃_{d} y

but

x ≃_{c} y .

Theorem5.2Characterization of semantic collapse

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

ρ_{d c, M} : L_{d} M ⟶ L_{c} M

has a homotopy fiber containing at least two inequivalent points belonging to the image of $η_{d} : M ⟶ L_{d} M$ .

Proof.

Suppose first that semantic collapse occurs. Then there exist $x, y \in M$ such that $η_{d} (x) \neq ≃ η_{d} (y)$ but $η_{c} (x) ≃ η_{c} (y)$ . Since

ρ_{d c, M} η_{d} (x) ≃ η_{c} (x)

and

ρ_{d c, M} η_{d} (y) ≃ η_{c} (y),

their images under $ρ_{d c, M}$ are equivalent. Thus $η_{d} (x)$ and $η_{d} (y)$ determine inequivalent points in a common homotopy fiber of $ρ_{d c, M}$ .

Conversely, suppose some homotopy fiber contains inequivalent points $u ≃ η_{d} (x)$ and $v ≃ η_{d} (y)$ coming from $M$ . Since $u$ and $v$ lie in the same homotopy fiber,

ρ_{d c, M} (u) ≃ ρ_{d c, M} (v) .

Hence $η_{c} (x) ≃ η_{c} (y)$ , while $η_{d} (x) \neq ≃ η_{d} (y)$ . Therefore semantic collapse occurs.

Corollary5.3Criterion for strict refinement

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

Example5.4Homotopical collapse

Suppose $M = S^{1}$ . At capability level $L_{0} = τ_{\leq 0}$ , one has

τ_{\leq 0} S^{1} ≃ *,

so every point becomes semantically indistinguishable. At capability level $L_{1} = τ_{\leq 1}$ , 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.

6Transmission

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

Definition6.1Transmission 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 ⟶ 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.

Definition6.2Replication witness space

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

Rep_{a \to b} (m) := Id_{L_{d} M} (T_{ab} (η_{c} (m)), η_{d} (m)),

where $c = κ (a)$ and $d = κ (b)$ .

Definition6.3Successful replication

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

T_{ab} (η_{c} (m)) ≃ η_{d} (m) .

Remark6.4Presentation 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 $η_{d} (m^{'}) ≃ η_{d} (m)$ . This formalizes the distinction between presentation identity and semantic identity.

7Replication 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_{b e} : L_{d} M \to L_{f} M .

Their composite is

T_{a e} := T_{b e} \circ T_{ab} .

Theorem7.1Composition of replication witnesses

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

Proof.

Let $p : T_{ab} (η_{c} (m)) = η_{d} (m)$ be a replication witness from $a$ to $b$ . Applying $T_{b e}$ gives

T_{b e} (p) : T_{b e} T_{ab} (η_{c} (m)) = T_{b e} (η_{d} (m)) .

Let $q : T_{b e} (η_{d} (m)) = η_{f} (m)$ be a replication witness from $b$ to $e$ . Path composition yields

q \circ T_{b e} (p) : T_{b e} T_{ab} (η_{c} (m)) = η_{f} (m) .

But $T_{b e} T_{ab} = T_{a e}$ . Therefore $T_{a e} (η_{c} (m)) = η_{f} (m)$ , so $Rep_{a \to e} (m)$ is inhabited.

Corollary7.2Iterated replication

For a chain of agents $a_{0} \to a_{1} \to \dots \to a_{n}$ , if $m$ replicates successfully across every edge, then $m$ replicates successfully under the composite transmission $a_{0} \to a_{n}$ .

8Mutation

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

Definition8.1Capability-relative mutation

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

presentation-preserving if $m^{'} ≃ m$ in $M$ ;
semantically faithful at $c$ if $m^{'} ≃_{c} m$ ;
a $c$ -detectable mutation if $m^{'} \neq ≃_{c} m$ .

Proposition8.2Mutation can disappear under coarse-graining

Let $c \leq d$ . It is possible that $m^{'} \neq ≃_{d} m$ while $m^{'} ≃_{c} m$ . 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.

Corollary8.3Capability-relative fidelity

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

Definition8.4Joint conservativity

The capability hierarchy is jointly conservative on $M$ if

(\forall c \in C, η_{c} (x) ≃ η_{c} (y)) ⟹ x ≃ y .

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

9Cognitive realization

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

Definition9.1Realization space

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

Real_{c} (m) := Map_{E_{c}} (L_{c} P_{m}, C_{c}) .

The meme is realizable at capability $c$ if $Real_{c} (m) \neq = \emptyset$ .

Definition9.2Capability-monotone realization system

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

Φ_{c d, m} : Real_{c} (m) ⟶ 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.

Theorem9.3Upward closure of realizability

In a capability-monotone realization system, $Real_{c} (m) \neq = \emptyset$ and $c \leq d$ imply $Real_{d} (m) \neq = \emptyset$ .

Proof.

Choose $r \in Real_{c} (m)$ . Applying $Φ_{c d, m}$ gives

Φ_{c d, m} (r) \in Real_{d} (m) .

Thus the latter realization space is inhabited.

Definition9.4Cognitive support

Define the cognitive support of $m$ by

Θ (m) := {c \in C ∣ Real_{c} (m) \neq = \emptyset} .

Corollary9.5Upward closure of support

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

c \in Θ (m), c \leq d ⟹ d \in Θ (m) .

Definition9.6Cognitive threshold

If $Θ (m)$ possesses a least element, define

χ (m) := min Θ (m) .

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

10Comprehension versus replication

Realization and replication are logically distinct.

Definition10.1Comprehension condition

An agent $b \in A_{d}$ can comprehend $m$ if $Real_{d} (m)$ is inhabited.

Definition10.2Replication condition

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

Proposition10.3Logical independence

Without additional assumptions,

Real_{d} (m) \neq = \emptyset

does not imply

Rep_{a \to b} (m) \neq = \emptyset,

and

Rep_{a \to b} (m) \neq = \emptyset

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 $Map (L_{d} P_{m}, C_{d})$ , whereas replication concerns the identity type $T_{ab} (η_{c} (m)) = η_{d} (m)$ . 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.

11Internal replication correspondence

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

R ⟶ A \times A \times M

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

Definition11.1Memetic replication correspondence

The span

A \times M s R 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 $(a, m) ⇝ (b, m^{'})$ .

Definition11.2Generalized replication correspondence

Define

R ⟶ A \times A \times M \times M

with fiber

R_{a, b, m, m^{'}} := Id_{L_{κ (b)} M} (T_{ab} (η_{κ (a)} m), η_{κ (b)} m^{'}) .

Then $(a, m) ⇝ (b, m^{'})$ 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 $m^{'} ≃_{κ (b)} m$ .

12Replication as a homotopy-coherent dynamical system

A sequence

(a_{0}, m_{0}) ⇝ (a_{1}, m_{1}) ⇝ \dots ⇝ (a_{n}, m_{n})

defines a path in the correspondence-generated dynamics. At each stage we may have $m_{i} \neq ≃ m_{i + 1}$ in the ambient object $M$ , even though $m_{i} ≃_{κ (a_{i + 1})} m_{i + 1}$ . 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.

Definition12.1Capability-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}} (η_{κ (a_{i})} m_{i}) = η_{κ (a_{i + 1})} m_{i + 1} .

Definition12.2Stable memetic lineage

Such a lineage is semantically stable relative to the recipients if

m_{i + 1} ≃_{κ (a_{i + 1})} m_{i}

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

13The central representation theorem

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

Theorem13.1Capability-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

η_{c} (m) ≃ η_{c} (m^{'})

in $L_{c} M$ . Equivalently, $m ≃_{c} m^{'}$ .

Proof.

By construction, all semantic states available at capability $c$ factor through the localization $η_{c} : M \to L_{c} M$ . Hence an agent at capability $c$ has access only to the images $η_{c} (m)$ and $η_{c} (m^{'})$ . 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 $L_{c} M$ .

Theorem13.2Replication preserves semantic identity

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

T_{ab} ([m]_{κ (a)}) ≃ [m]_{κ (b)} .

Proof.

By Definition 6.2,

Rep_{a \to b} (m) = Id_{L_{κ (b)} M} (T_{ab} ([m]_{κ (a)}), [m]_{κ (b)}) .

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

14Apparent fidelity and cognitive coarseness

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

Theorem14.1Coarse observers weakly increase apparent fidelity

Let $c \leq d$ . 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 $m^{'} ≃_{d} m$ . By Theorem 4.1, $m^{'} ≃_{c} m$ . The converse fails whenever semantic collapse occurs between $d$ and $c$ , by Theorem 5.2.

Corollary14.2Measured 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.

Remark14.3Sloganization 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.

15Memetic complexity

Capability thresholds induce a natural notion of semantic complexity.

Definition15.1Capability complexity

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

Comp (m) := χ (m) .

Thus $Comp (m)$ is the least capability required to realize $m$ .

Proposition15.2Complexity monotonicity under realization

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

Proof.

Immediate from Theorem 9.3.

16Semantic compression

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

ρ_{d c, M} : L_{d} M \to L_{c} M

forgets distinctions inaccessible at level $c$ .

Definition16.1Capability-relative compression

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

Proposition16.2Fibers detect collapsed distinctions

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

Proof.

This is the fiberwise form of Theorem 5.2.

17Information loss as failure of conservativity

A transmission may destroy distinctions.

Definition17.1Semantically conservative transmission

A transmission $T : L_{c} M \to L_{d} M$ is semantically conservative if it reflects equivalences:

T (x) ≃ T (y) ⟹ x ≃ y .

Definition17.2Semantic information loss

Transmission exhibits semantic information loss if there exist

x \neq ≃ y

such that

T (x) ≃ T (y) .

Proposition17.3Conservativity 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.

18Replication, compression, and selection

The foregoing structures distinguish three independent components.

Realizability

Real_{c} (m)

Transmissibility

T_{ab}

Semantic fidelity

Rep_{a \to b} (m)

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

effective replication rate \sim exposure \times realizability \times transmission probability \times capability-relative fidelity,

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

19Formal specification

The complete deterministic specification may be summarized by the tuple

M = (E, C, {L_{c}}_{c \in C}, {A_{c}}_{c \in C}, M, {C_{c}}_{c \in C}, {T_{ab}}) .

Symbol	Meaning
$E$	Ambient ∞-topos.
$C$	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 ≃_{c} m^{'} ⟺ Id_{L_{c} M} (η_{c} (m), η_{c} (m^{'})) is inhabited.

The fundamental replication criterion is

Rep_{a \to b} (m) = Id_{L_{κ (b)} M} (T_{ab} (η_{κ (a)} m), η_{κ (b)} m) .

20Main structural consequences

The preceding results may be summarized as follows.

Theorem A — Equivalence. For each capability $c$ , the relation $≃_{c}$ is an equivalence relation.
Theorem B — Capability monotonicity. If $c \leq d$ , then $x ≃_{d} y$ implies $x ≃_{c} y$ .
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.

21Interpretation

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} = η_{c} (m) \in L_{c} M .

Hence memetic identity is indexed by capability:

m ⟼ {[m]_{c}}_{c \in C} .

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

T_{ab} ([m]_{κ (a)}) ≃ [m]_{κ (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 $c \leq d$ induces the refinement

[m]_{d} \mapsto [m]_{c},

and consequently

m ≃_{d} m^{'} ⟹ m ≃_{c} m^{'}, m ≃_{c} m^{'} does not imply m ≃_{d} m^{'} .

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

22Conclusion

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 $E$ , a hierarchy of left-exact localizations $L_{c} : E \to E_{c}$ , and a memetic state object $M$ , the identity of a meme for an observer of capability $c$ is represented by its localized state $η_{c} (m) \in L_{c} M$ .

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

T_{ab} (η_{κ (a)} m) ≃ η_{κ (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.