Evolution of the semantic universe

Abstract

We extend capability-relative memetic replication beyond evolution inside a fixed cognitive site. The category of cognitive architectures, its translation morphisms, and its Grothendieck topology are now themselves dynamical.

At time $t$, let

be a cognitive ∞-site and let

be its semantic ∞-topos. A transition of semantic universes is a left-exact colimit-preserving functor

or, more generally, a correspondence of ∞-topoi. Memetic objects are sections of an indexed family of semantic universes rather than objects of one fixed ∞-topos.

We distinguish four phenomena: change of site presentation without semantic change; refinement of the Grothendieck topology; creation or deletion of cognitive architectures and translations; and genuine change of the semantic ∞-topos. Left-exact transport preserves identity witnesses, conservative transport preserves and reflects semantic identity, refinement of a topology induces a left-exact localization, and equivalent site presentations yield equivalent semantic theories.

Stochastic evolution across changing semantic spaces uses a variable-space mutation kernel

and the non-autonomous replicator–mutator equation

Finally, the semantic universe itself may be selected or transformed by its resident memes. A total state is a point of a Grothendieck construction over an ∞-category of admissible semantic universes. The resulting hierarchy is

The preceding fixed-universe theory is Note VI.

Notation

$({\mathcal{C}}_t,J_t)$

The cognitive site at stage t. Objects, translations, and covers may all change.

${\mathcal{E}}_t$

The semantic ∞-topos of that site.

$F_{t,t^{\prime }}$

Left-exact transport of semantic objects from one universe into the next.

$K_t$

A mutation kernel whose target lives in the next semantic space.

$\mathfrak{U}$

The ∞-category of admissible semantic universes.

$(u,s)$

A universe together with a semantic state inside it.

1Why the site must itself become dynamical

In the preceding theory, one fixed cognitive site $(\mathcal{C},J)$ determined one semantic universe ${\mathrm{Sh}}_{\infty }(\mathcal{C},J)$. Cognitive architectures could evolve inside $\mathcal{C}$, but three things were held fixed: the objects of $\mathcal{C}$, the admissible translations ${\mathrm{Map}}_{\mathcal{C}}(c,d)$, and the Grothendieck topology $J$. We now allow all three to vary. At time $t$, write

Semantic evolution can alter objects, morphisms, covers, and sheaves, and therefore the ambient notion of semantic object itself.

2Time as an indexing ∞-category

Let $\mathcal{T}$ be an ∞-category of times or evolutionary stages. For discrete time one may take $\mathcal{T}=\mathbb{N}$, regarded as a category with a unique arrow $n\to m$ whenever $n\le m$. More general choices permit branching timelines, partially ordered stages, or higher homotopies between evolutionary histories.

3Dynamical cognitive sites

4Invariant semantic-universe transitions

Because a single ∞-topos can have many inequivalent-looking site presentations, the evolution of sites should be formulated at the level of ∞-topoi.

5Dynamic semantic objects

A memetic object can no longer be one fixed $M\in \mathcal{E}$. It must vary with the semantic universe.

6Identity transport

Let $x,y:1\to X$ be points of $X\in {\mathcal{E}}_t$. Their identity object is the pullback

along the diagonal.

7Conservative evolution

8Presentation change versus genuine universe change

A different site presentation does not necessarily produce a different semantic universe.

9Evolution of the Grothendieck topology

One way to change a semantic universe without changing the architecture category is to modify which families count as covers. Let $J\subseteq J^{\prime }$ be Grothendieck topologies on the same ∞-category $\mathcal{C}$, with $J^{\prime }$ finer than $J$.

10Interpretation of topology refinement

Adding covers imposes additional local-to-global compatibility equations. Thus

produces

A semantic object that fails the new descent conditions is replaced by its closest $J^{\prime }$-local approximation. Evolution of $J$ can change which distributed representations count as legitimate global semantic objects.

11Descent constraints as semantic laws

Suppose $\{U_i\to U\}$ becomes a cover only after the transition $J⇝J^{\prime }$. Then a formerly unconstrained semantic object $X$ must after refinement satisfy

Evolution of the topology can create new equations among local cognitive perspectives.

12Creation of new architectures

Changing $\mathcal{C}$ modifies the possible observer types. Suppose $i:{\mathcal{C}}_t\to {\mathcal{C}}_{t+1}$ describes surviving architectures.

13New translation morphisms

Even if the object set does not materially change, new morphisms may appear. Suppose $c$ and $d$ persist. If

fails to be essentially surjective, then new translation procedures exist at time $t+1$. Semantic-universe evolution can create new architectures, or merely new ways of translating among old ones. The latter matters because descent depends on morphisms, not only objects.

14Why ordinary functors do not model deletion symmetrically

A forward functor ${\mathcal{C}}_t\to {\mathcal{C}}_{t+1}$ models persistence and the creation of new target objects, but it does not symmetrically encode architectures that simply cease to participate. The more general formalism is a correspondence.

15Site-transition correspondences

16Semantic-universe correspondences

The same construction lifts from sites to their ∞-topoi. A general semantic transition may be

This is an object of a correspondence $(\infty ,2)$-category of ∞-topoi. Such correspondences are useful when there is no preferred global functor ${\mathcal{E}}_t\to {\mathcal{E}}_{t+1}$. They represent partial translation, shared semantic subuniverses, or many-to-many reconstruction.

17Composition of semantic correspondences

Given spans through ${\mathcal{E}}_1$, their composite is the homotopy pullback

This produces a span from ${\mathcal{E}}_0$ to ${\mathcal{E}}_2$. Associativity holds up to the coherent equivalence of homotopy pullbacks. Semantic-universe histories live in a higher category of correspondences.

18Dynamic descent

Suppose $M_t\in {\mathcal{E}}_t$ satisfies descent for a cover $U_{\bullet ,t}\to U_t$, and $F_{t,t^{\prime }}$ is left exact and colimit-preserving. If the image diagram remains a cover, descent transports.

19Semantic novelty

20Novel distinctions

A later universe may possess observables unavailable earlier. Let $X,Y\in {\mathcal{E}}_t$. Suppose their images are distinguished by a new observable

that does not descend from any corresponding observable at time $t$. The later universe has acquired a new semantic distinction: a structural increase in expressive power, not merely a new data value.

21Semantic collapse

22Semantic phase transition

23Dynamic identity

Let $M_t\in {\mathcal{E}}_t$ be a dynamical memetic object and let $x_t,y_t\in M_t$. The identity space ${\mathrm{Id}}_{M_t}(x_t,y_t)$ now varies in a varying ∞-topos.

24Variable semantic state spaces

Let $S_t\in {\mathcal{E}}_t$ be the observer-relative semantic state object at time $t$. A population is

where the probability structure may now vary with the semantic universe.

25Compatible probability structure

26Mutation across different semantic universes

27Variable-space selection

Let $w_t:S_t\to {\mathbb{R}}_{>0}$ be the current fitness field. Define

Selection yields

28Non-autonomous replicator–mutator equation

29Non-autonomous transfer operators

Define

Then ${\mathcal{L}}_t$ maps positive measures on $S_t$ to positive measures on $S_{t+1}$. The normalized update is

There is no single fixed transfer operator. One has a cocycle ${\mathcal{L}}_{t+n-1}\cdots {\mathcal{L}}_t$.

30Iterated variable-universe evolution

31No fixed equilibrium in general

When $S_t$ changes with time, the equation ${\nu }_{\ast }=\mathcal{R}({\nu }_{\ast })$ is not even type-correct unless the semantic universes repeat or are canonically identified. The appropriate invariant is not necessarily a fixed point.

32The space of semantic universes

Semantic-universe evolution can itself be endogenous. Let $\mathfrak{U}$ be an ∞-category of admissible cognitive semantic universes. A point $u\in \mathfrak{U}$ may encode $({\mathcal{C}}_u,J_u,{\mathcal{E}}_u)$ together with the probability, modality, and memetic structures required.

33Indexed semantic universes

Suppose we have a functor

with $u\mapsto {\mathcal{E}}_u$. Choose in each universe a semantic state object $S_u\in {\mathcal{E}}_u$. These form an indexed semantic family.

34Total universe–semantic state space

Straightening determines a total ∞-category ${\int }_{\mathfrak{U}}S$. Define

A point is a pair $(u,s)$, a semantic universe together with a semantic state inside that universe.

35Universe projection

There is a projection

Every population on $\Omega$ has a universe marginal ${\pi }_{\mathfrak{U}\ast }\nu$. The system can contain a distribution over different semantic universes.

36Universe-changing kernels

37Full universe–semantic kernel

The invariant dynamics is represented by

Its universe marginal is $\mathbb{P}({\pi }_{\mathfrak{U}})\circ K$. One transition can simultaneously change $u\mapsto u^{\prime }$ and $s\mapsto s^{\prime }\in S_{u^{\prime }}$.

38Universe-dependent fitness

Let

Then $w((u,s),\nu ,e)$ measures the reproductive value of state $s$ as represented inside universe $u$.

39Semantic-universe coevolution operator

Define

Then

This is formally the joint replicator–mutator operator, but the state coordinate now includes the semantic universe itself.

40Universe selection

The marginal distribution on semantic universes evolves according to

Semantic universes can spread, disappear, or transform under selection and transition.

41Memes that alter their semantic universe

42Strong semantic reflexivity

43Evolution of the coverage relation

A particularly important universe-reflexive transition changes only $J_u$ while leaving the underlying architecture category fixed. Suppose $J_u⊊J_{u^{\prime }}$. Then ${\mathcal{E}}_{u^{\prime }}$ is a left-exact localization of ${\mathcal{E}}_u$. A resident semantic population can therefore induce new descent requirements.

44Evolution of translation structure

Another transition changes ${\mathrm{Map}}_{{\mathcal{C}}_u}(c,d)$. A newly created translation may cause previously disconnected semantic regions to participate in common descent diagrams. New communication capability can change global semantic structure even when local observer objects remain unchanged.

45Emergent global semantic objects

Suppose at time $t$ the architectures $c_1$ and $c_2$ have no relevant common cover. At $t+1$, new morphisms produce a cover

Compatible local states $x_1\in M(c_1)$ and $x_2\in M(c_2)$ may now descend to a global object $x\in M(c)$. Evolution of the site can make a new distributed semantic object globally definable.

46Loss of global semantic objects

Conversely, changes in topology or architecture may destroy the compatibility conditions required for a previous global representation. A formerly meaningful global section may cease to admit transport into the new semantic universe. Universe evolution can produce both semantic emergence and semantic extinction.

47Semantic extinction

48Creation of semantic dimensions

A new architecture or observable may introduce a semantic coordinate unavailable at earlier times. Model this by a map $S_{t+1}\to S_t$ whose homotopy fibers are noncontractible. Then multiple new semantic states project to one old state.

49Creation of new homotopy structure

The refinement need not occur only at the level of connected components. A semantic state may acquire nontrivial ${\pi }_1$, ${\pi }_2$, and higher information. Universe evolution can alter the relevant homotopy $n$-type of semantic objects, gaining or losing higher coherence.

50Dynamic Postnikov resolution

A simple example is a time-dependent truncation level $n_t$. Let ${\mathcal{E}}_t$ retain semantic information only through ${\tau }_{\le n_t}$. If $n_{t+1}>n_t$, new higher identity structure becomes visible. If $n_{t+1}<n_t$, previously visible higher distinctions collapse. The capability hierarchy is a one-dimensional special case of evolving semantic universes.

51Path dependence

When the universe itself changes, the final semantic state may depend on the path. Suppose one history is $u_0\to u_1\to u_3$ and another is $u_0\to u_2\to u_3$. The composites

need not be equivalent unless the evolution diagram contains a specified coherent 2-cell between them. Semantic evolution is generally path dependent.

52Coherent path independence

53Hysteresis

If different paths through semantic-universe space produce non-equivalent transports between the same endpoints, the system exhibits semantic hysteresis:

for two paths $\gamma ,{\gamma }^{\prime }$ from a common start to a common end $v$. Semantic history cannot be reconstructed from the endpoint universe alone.

54Universe-level hidden state

If an observer sees only a coarse semantic object $s_t$ but not the current semantic universe $u_t$, the observed dynamics can be non-Markovian even when $(u_t,s_t)$ is Markovian. The semantic universe can itself function as latent state.

55Universe lumpability

Let $q:\Omega \to Z$ be a coarse observable that forgets some or all universe information.

56Universe-level Price equation

Let $\varphi :\Omega \to \mathbb{R}$ be an observable of the semantic universe, such as a numerical invariant of complexity or of the number of architecture types. Then

The first term measures selection among semantic universes. The second measures direct transformation of semantic universes within lineages.

57Endogenous generation of semantic universes

The kernel $K$ may assign positive probability to universe states $u^{\prime }$ not previously represented in the support of the population. Then

Genuinely new semantic universes can enter the evolutionary process.

58Semantic innovation rate

Given the current support $U_t=\mathrm{supp}(a_t)$, the probability of semantic-universe innovation is

In continuous settings this requires a measurable notion of previous support. The quantity measures population flow into previously unoccupied semantic universes.

59Fixed-universe theory as a special case

Suppose the universe transition kernel is ${\delta }_u$. Then $u_t=u_0$ almost surely, and the theory reduces to memetic and cognitive evolution inside the fixed ∞-topos ${\mathcal{E}}_{u_0}$.

60Fixed-site but evolving architecture theory

If the semantic universe $u$ is fixed but the state object includes an architecture coordinate $c\in {\mathcal{C}}_u$, we recover the cognitive–memetic coevolution theory of Note VI. The inclusions of generality are

61Four distinct notions of change

The theory now distinguishes four categorically distinct phenomena.

1. State change. $m\to m^{\prime }$ inside one semantic state space.
2. Observer change. $c\to c^{\prime }$ inside one architecture category.
3. Semantic-rule change. $J\to J^{\prime }$, or a change of translation morphisms.
4. Universe change. $\mathcal{E}\to {\mathcal{E}}^{\prime }$ with $\mathcal{E}\not\simeq {\mathcal{E}}^{\prime }$.

62Main semantic-universe evolution theorem

63Main evolutionary theorem

64Main reflexivity theorem

65Total hierarchy

The complete formal hierarchy is now

Evolution can occur at every level.

66Final principle

In the fixed-universe theory the question was whether a transformation preserves a meme. In the observer-relative theory it became whether it preserves the meme relative to a cognitive modality. In the coevolutionary theory it became how the meme alters the modality that later judges its identity. The present theory asks how the evolving population alters the entire semantic universe in which such questions can be formulated.

The causal chain is semantic state, architecture change, site change, new descent structure, a new ∞-topos, new identity types, and then new replication dynamics.

Memetic evolution can modify the universe of semantic possibility itself: dynamics of states, of observers, of descent, and of the ambient ∞-topos. In compact form,

where even the type of the state being evolved is permitted to change.