Essay I

An (∞)-Topos-Theoretic Separation of Organisms, Viruses, and Computers

Autonomous closure, host-relative closure, and externally maintained machinery are distinct, and the functors that identify a virus with a program do not preserve the distinction.

Contents28 sections

Abstract

We formulate a categorical distinction between living organisms, viruses, and computational machines inside an $(\infty)$ -topos $E$ . The distinction is not identified with computational behavior, nor with material substrate alone. A physical system carries typed components, internally generated processes, required organizational resources, and interfaces to an environment.

For each organizational mode $a$ , a system $X$ has a demand morphism and an endogenous production morphism

d_{a}^{X} : D_{a} (X) ⟶ R_{a} (X), p_{a}^{X} : P_{a} (X) ⟶ R_{a} (X) .

The system is closed in mode $a$ when $im (d_{a}^{X}) \leq im (p_{a}^{X})$ in the subobject lattice of $R_{a} (X)$ . Everything required as an organizational enabling condition is then generated by processes internal to the system. Matter and energy may still enter through explicitly external channels.

This yields three notions: autonomous closure, host-relative closure, and externally maintained machinery. A cellular organism is modeled as organizationally closed over an appropriate signature. A virus is not autonomously closed, but becomes closed after supplementation by host processes. An ordinary computer is not closed for fabrication, repair, and physical reproduction, though it may be closed for a restricted signature of computational operations.

These closure properties are homotopy invariant, preserved by equivalences and by geometric inverse-image functors, and satisfy descent. They are therefore genuine obstructions to equivalence. A material-support invariant is defined separately and shown to be logically independent of organizational closure. The analogy between a virus and a program arises only after a non-conservative forgetful functor erases structure needed to compare either of them with a cell.

1Ambient (∞)-topos

Let $E$ be an $(\infty)$ -topos. One may think of $E$ as a category of homotopy-coherent sheaves of physical states, processes, or observations, over a site of spacetime regions, experimental contexts, or interaction contexts.

An $(\infty)$ -topos, rather than an ordinary category, supplies the following.

arbitrary higher homotopies between realizations descent and locality an internal intuitionistic higher-order logic effective epimorphism–monomorphism factorizations a well-behaved theory of subobjects and images

The properties used below are the standard ones of $(\infty)$ -topoi in the sense of Higher Topos Theory.

Definition 1.1

Subobject

For $Y \in E$ , a subobject of $Y$ is a $(- 1)$ -truncated morphism $A ↪ Y$ . Equivalence classes of such morphisms form a partially ordered set $Sub_{E} (Y)$ . The order is $A \leq B$ if and only if $A ↪ Y$ factors through $B ↪ Y$ . Internally, $A \leq B$ means that every element satisfying $A$ also satisfies $B$ .

Definition 1.2

Image

Every morphism $f : X \to Y$ admits a canonical factorization

X ↠ im (f) ↪ Y,

in which the first factor is $(- 1)$ -connected — equivalently, an effective epimorphism — and the second is $(- 1)$ -truncated. This is the $(- 1)$ -connected/ $(- 1)$ -truncated factorization system of an $(\infty)$ -topos. Hence $im (f) \in Sub (Y)$ is well defined up to a contractible space of choices.

2Typed physical systems

We first encode material composition. Let $K \in E$ be an object of physical component types. In a sufficiently concrete model, points of $K$ might include protein, lipid, DNA, silicon, copper, transistor, ribosome, and so on. No discreteness assumption is required.

Definition 2.1

Typed system

A $K$ -typed physical system $X$ consists in part of an object $C_{X} \in E$ of components together with a typing morphism $τ_{X} : C_{X} \to K$ . Its material support is the image

Supp (X) := im (τ_{X}) ↪ K .

Thus $Supp (X)$ is the subobject of component-types actually instantiated by $X$ .

Definition 2.2

Type-preserving equivalence

An equivalence $X ≃ Y$ of $K$ -typed systems is required to contain an equivalence $f : C_{X} \sim C_{Y}$ and a homotopy-commutative square

C_{X} τ_{X} ↓ ⏐ K f C_{Y} ↓ ⏐ τ_{Y} K,

together with the analogous coherent equivalences on the remaining structure. Commutativity says $τ_{X} ≃ τ_{Y} \circ f$ .

3The substrate invariant

Theorem 3.1

Material-support invariance

If $X ≃ Y$ as $K$ -typed physical systems, then $Supp (X) ≃ Supp (Y)$ in $Sub (K)$ .

Proof. The equivalence supplies $f : C_{X} \sim C_{Y}$ with $τ_{X} ≃ τ_{Y} \circ f$ . Homotopic morphisms determine the same image subobject: image depends on the morphism only through its class in the homotopy category, via the canonical $(- 1)$ -connected/ $(- 1)$ -truncated factorization. Precomposition with an equivalence does not change that image, so

im (τ_{X}) ≃ im (τ_{Y} \circ f) ≃ im (τ_{Y}) .

Hence $Supp (X) ≃ Supp (Y)$ . The general homotopy statement is Lemma 6.1.

End of proof.

Corollary 3.2

Substrate obstruction

If $Supp (X) \neq ≃ Supp (Y)$ , there is no equivalence $X ≃ Y$ in any category whose equivalences preserve the physical typing $τ \to K$ .

Differences of physical composition are therefore a genuine obstruction, provided composition is among the structure being preserved. A category that deliberately forgets $τ$ cannot, by construction, detect the distinction.

4Organizational signatures

Material composition is not the only relevant invariant. Let $Σ$ be a collection of organizational modes. Typical elements might be

Σ = {boundary, catalysis, repair, translation, replication, regulation, \dots} .

The claim is not that every organism instantiates exactly this list. The signature $Σ$ records which structural capacities a given theory treats as organizationally essential.

Generic matter and free energy need not belong to $Σ$ . Living systems are thermodynamically open. The closure defined here is not closure to matter and energy. It is closer to Rosen’s closure to efficient causation, and to the organizational closure of Maturana and Varela. Maturana characterized autopoietic organization as a network of component-producing processes that recursively regenerates the network and its boundary. Rosen’s relational biology likewise separates material flows from internally regenerated organizational causes.

5Organizational resource data

Definition 5.1

Σ-system

A $Σ$ -system $X$ consists, for each $a \in Σ$ , of an organizational-resource object $R_{a} (X)$ , a demand morphism

d_{a}^{X} : D_{a} (X) ⟶ R_{a} (X),

and an endogenous-production morphism

p_{a}^{X} : P_{a} (X) ⟶ R_{a} (X) .

Here $D_{a} (X)$ enumerates organizational enablers required by $X$ , and $P_{a} (X)$ enumerates organizational enablers produced by processes belonging to $X$ . Write

Req_{a} (X) := im (d_{a}^{X}), End_{a} (X) := im (p_{a}^{X}) .

Both are subobjects of $R_{a} (X)$ .

Definition 5.2

Closure in a mode

The system $X$ is closed in mode $a$ when $Req_{a} (X) \leq End_{a} (X)$ . Equivalently, the demand image factors through the production image:

Req_{a} (X) ⟶ End_{a} (X) ↪ R_{a} (X) .

Write $Cl_{a} (X)$ for this proposition.

Definition 5.3

Σ-closure

The system is $Σ$ -closed when it is closed in every mode:

Cl_{Σ} (X) ⟺ a \in Σ ⋀ Cl_{a} (X) ⟺ a \in Σ ⋀ (Req_{a} (X) \leq End_{a} (X)) .

This is an internal proposition of the theory.

6Homotopy invariance

Lemma 6.1

Homotopic maps have the same image

If $d ≃ d^{'} : D \to R$ , then $im (d) ≃ im (d^{'})$ . The same holds for production maps.

Proof. Image is determined by the canonical $(- 1)$ -connected/ $(- 1)$ -truncated factorization of the morphism represented in the homotopy category. Homotopic morphisms define equivalent objects of the arrow $(\infty)$ -category $Fun (Δ^{1}, E)$ , and therefore equivalent image factorizations. Their image subobjects coincide.

End of proof.

Corollary 6.2

Closure is not presentation-dependent

The truth value of $Cl_{a} (X)$ is invariant under all coherent higher homotopies in the presentation of $X$ . The closure invariant is $(\infty)$ -categorical.

7Closure is preserved by equivalence

Theorem 7.1

Organizational closure is invariant

Let $X ≃ Y$ be an equivalence of $Σ$ -systems preserving the organizational-resource diagrams. Then $Cl_{a} (X)$ if and only if $Cl_{a} (Y)$ , for every $a \in Σ$ . Consequently $Cl_{Σ} (X)$ if and only if $Cl_{Σ} (Y)$ .

Proof. A structure-preserving equivalence supplies $R_{a} (X) ≃ R_{a} (Y)$ , $D_{a} (X) ≃ D_{a} (Y)$ , and $P_{a} (X) ≃ P_{a} (Y)$ , under which the demand and production diagrams agree up to coherent homotopy. By Lemma 6.1 the image subobjects correspond:

Req_{a} (X) ≃ Req_{a} (Y), End_{a} (X) ≃ End_{a} (Y) .

An equivalence induces an isomorphism of the corresponding subobject posets and preserves their order. Therefore $Req_{a} (X) \leq End_{a} (X)$ if and only if $Req_{a} (Y) \leq End_{a} (Y)$ .

End of proof.

Corollary 7.2

Organizational obstruction

If $Cl_{a} (X)$ and $\neg Cl_{a} (Y)$ for some mode $a$ , then $X \neq ≃ Y$ as $Σ$ -systems.

This is the basic separation theorem. Equivalence cannot cross a difference of closure.

8External supplementation

A virus is not handled by merely changing the context in which it is observed. The host contributes machinery. Base change and supplementation are different operations.

Suppose an external system $H$ supplies additional $a$ -resources along $q_{a}^{H} : Q_{a} (H) \to R_{a} (X)$ . Form the combined supply

P_{a} (X) ⊔ Q_{a} (H) [p_{a}^{X}, q_{a}^{H}] R_{a} (X) .

Definition 8.1

Relative closure

The system $X$ is closed in mode $a$ relative to $H$ when

Req_{a} (X) \leq im ([p_{a}^{X}, q_{a}^{H}]) .

Write $Cl_{a} (X ∣ H)$ .

Definition 8.2

Strict external dependence

The system $X$ is strictly dependent on $H$ in mode $a$ when $\neg Cl_{a} (X)$ but $Cl_{a} (X ∣ H)$ . Thus $X$ cannot generate all of its required organizational resources, while the coupled system can.

9Image of combined production

Lemma 9.1

Image of a copairing

Let $p : P \to R$ and $q : Q \to R$ . Then

im ([p, q]) = im (p) \lor im (q)

in $Sub (R)$ .

Proof. Both $p$ and $q$ factor through $im ([p, q])$ , so $im (p) \leq im ([p, q])$ and $im (q) \leq im ([p, q])$ . Hence their join is contained in the image of the copairing.

Conversely, $[p, q]$ factors through every subobject through which both $p$ and $q$ factor, and in particular through $im (p) \lor im (q)$ . By minimality of the image, $im ([p, q]) \leq im (p) \lor im (q)$ .

End of proof.

Relative closure may therefore be rewritten

Req_{a} (X) \leq End_{a} (X) \lor Host_{a} (H),

where $Host_{a} (H) := im (q_{a}^{H})$ .

10Base change and descent

Let $f^{*} : E \to F$ be the inverse-image functor of a geometric morphism of $(\infty)$ -topoi. It preserves finite limits and all small colimits.

Lemma 10.1

Inverse image preserves factorization

The functor $f^{*}$ preserves effective epimorphisms, monomorphisms, and image factorizations.

Proof. Effective epimorphisms are the morphisms whose Čech nerves are colimit diagrams. Since $f^{*}$ preserves colimits and finite limits, it sends Čech nerves to Čech nerves, and effective epimorphisms to effective epimorphisms. Left exactness preserves $(- 1)$ -truncated morphisms, hence monomorphisms. Applying $f^{*}$ to an effective-epimorphism/monomorphism factorization therefore yields the image factorization after base change.

End of proof.

Theorem 10.2

Geometric invariance

If $Cl_{a} (X)$ holds in $E$ , then $Cl_{a} (f^{*} X)$ holds in $F$ .

Proof. From $Req_{a} (X) \leq End_{a} (X)$ , apply $f^{*}$ . Lemma 10.1 gives

f^{*} Req_{a} (X) ≃ Req_{a} (f^{*} X)

and the analogous identification for endogenous production. Since $f^{*}$ preserves factorization through monomorphisms,

Req_{a} (f^{*} X) \leq End_{a} (f^{*} X) .

End of proof.

11Closure satisfies descent

Let $u : U ↠ B$ be an effective epimorphism. Pullback induces $u^{*} : Sub (B) \to Sub (U)$ . Along an effective epimorphism this map is conservative on subobjects: it reflects order.

Theorem 11.1

Local-to-global closure

Suppose a system over $B$ becomes organizationally closed after pullback along an effective epimorphism $u : U ↠ B$ . Then it was already organizationally closed over $B$ .

Proof. The hypothesis is $u^{*} Req_{a} (X) \leq u^{*} End_{a} (X)$ . Pullback along an effective epimorphism reflects the order of subobjects, so $Req_{a} (X) \leq End_{a} (X)$ . Thus $Cl_{a} (X)$ . Applying this to every $a \in Σ$ yields $Cl_{Σ} (X)$ .

End of proof.

Corollary 11.2

A host is not a local context

Suppose $\neg Cl_{a} (V)$ . There is no effective-epimorphic observational cover $U ↠ 1$ such that mere restriction to $U$ makes $V$ closed. If $Cl_{a} (u^{*} V)$ held, Theorem 11.1 would give $Cl_{a} (V)$ .

Host dependence cannot be represented by localization or by a change of observational context. The host must contribute additional organizational morphisms $q_{a}^{H} : Q_{a} (H) \to R_{a} (V)$ . Restriction and coupling are different.

12Axiomatized cells

Let $Σ_{org}$ contain modes such as boundary, catalysis, repair, biosynthesis, and replication.

Definition 12.1

Autopoietic Σ-object

An autopoietic $Σ$ -object is a system $B$ with $Cl_{Σ_{org}} (B)$ . External material and energy maps $M_{ext} \to B$ are explicitly permitted. The condition says only that the organizational machinery named by $Σ_{org}$ is internally regenerated.

This is not thermodynamic closure. It formalizes the core of autopoiesis and of closure to efficient causation. A single-celled organism is then modeled as an object

B \in Aut_{Σ} (E) \subseteq Sys_{Σ} (E) .

That is an empirical modeling hypothesis, not a theorem of pure mathematics.

13Axiomatized viruses

Viral replication depends on host-cell machinery: on host metabolism, translation, membranes, and biosynthetic resources. Let $V$ be a viral system and $H$ a host.

Definition 13.1

Host-dependent replicator

The virus $V$ is a host-dependent replicator if for some $a \in Σ_{rep}$ ,

\neg Cl_{a} (V) and Cl_{a} (V ∣ H) .

Equivalently, $Req_{a} (V) ≰ End_{a} (V)$ , while

Req_{a} (V) \leq End_{a} (V) \lor Host_{a} (H) .

14Axiomatized conventional computers

Let $C$ be an ordinary contemporary computer. Several signatures are available. For the restricted signature of instruction execution, one may have $Cl_{Σ_{exec}} (C)$ : a running machine can carry out many of the operations required for continued execution using its own hardware.

Enlarge the signature to repair, chip fabrication, component replacement, assembly, and physical reproduction. For an ordinary computer, $\neg Cl_{Σ_{phys}} (C)$ . There is instead an industrial environment $I$ such that, for the relevant modes,

Req_{a} (C) \leq End_{a} (C) \lor Ind_{a} (I),

so that $Cl_{a} (C ∣ I)$ may hold while $Cl_{a} (C)$ does not.

The causal picture is then approximately $I ⟶ C$ , where $I$ contains semiconductor fabrication, mining, refining, electricity generation, human maintenance, transport, and manufacturing infrastructure.

15Cell/computer separation

Theorem 15.1

Cell and computer

Let $B$ be an axiomatized cellular organism and $C$ an ordinary computer. Suppose some mode $a$ satisfies $Cl_{a} (B)$ and $\neg Cl_{a} (C)$ . Then $B \neq ≃ C$ as $Σ$ -systems.

Proof. An equivalence $B ≃ C$ would imply $Cl_{a} (B)$ if and only if $Cl_{a} (C)$ , by Theorem 7.1.

End of proof.

16Virus/cell separation

Theorem 16.1

Virus and cell

Suppose $B$ is autonomous in a mode $a$ , so $Cl_{a} (B)$ , while a virus $V$ is strictly host-dependent in that mode: $\neg Cl_{a} (V)$ , though $Cl_{a} (V ∣ H)$ for some host. Then $V \neq ≃ B$ .

Proof. Immediate from Theorem 7.1. Relative closure of $V$ with $H$ is a property of a different structured system, not of $V$ alone.

End of proof.

A virus may be biologically adjacent to life without inhabiting the same full subcategory as autonomously organized cells.

17The virus/computer analogy

There is nevertheless a legitimate abstraction under which viruses and programs resemble one another. Let $Sys_{Σ}$ be the category of structured physical systems, and let

U : Sys_{Σ} ⟶ Dep

be the forgetful functor that remembers only a replicator or executable specification, a required host, and successful production after host supplementation. It forgets material typing, biochemical organization, evolutionary lineage, boundary production, thermodynamic organization, and most of the physical causal structure.

Under $U$ one may have $U (V) ≃ U (P)$ for a virus $V$ and a program $P$ . Both have the shape “replicator plus host yields copies”:

V + H ⟶ V^{⊔ n}, P + M ⟶ P^{⊔ n} .

18Non-conservativity of the analogy

Proposition 18.1

The dependency functor need not be conservative

The functor $U : Sys_{Σ} \to Dep$ need not reflect equivalences. That is, $U (X) ≃ U (Y)$ does not imply $X ≃ Y$ .

Proof. Choose $X$ and $Y$ with $U (X) ≃ U (Y)$ but $Supp (X) \neq ≃ Supp (Y)$ . Theorem 3.1 forbids $X ≃ Y$ . Since $U$ forgets material support, it may still identify their dependency skeletons.

End of proof.

Corollary 18.2

Analogy is not equivalence

The statement that viruses are like computer programs can be correct after a sufficiently coarse forgetful functor, without entailing that a virus is equivalent to a computer as a physical system, and still less that a computer is equivalent to a cell.

19The original argument, categorically

Let $B$ be a bacterium, $V$ a virus, and $C$ a computer. At the coarse dependency level one may have $U (V) ≃ U (C)$ . In the richer physical category, $V \neq ≃ C$ as soon as an invariant differs, and $B \neq ≃ V$ when autonomous closure differs. The inference

U (V) ≃ U (C) ⟹ B ≃ C

is invalid. Equality after $U$ is equality in the quotient semantics $U$ induces. It says nothing about the domain unless $U$ is conservative. Since $U$ is not conservative, the inference fails.

20The combined invariant

Define

I (X) = (Supp (X), {Cl_{a} (X)}_{a \in Σ}) .

This records material support together with the closure profile.

Theorem 20.1

The combined invariant is preserved

If $X ≃ Y$ as fully structured physical systems, then $I (X) = I (Y)$ .

Proof. Material support is invariant by Theorem 3.1. Each closure component is invariant by Theorem 7.1.

End of proof.

Corollary 20.2

Two independent obstructions

If $Supp (X) \neq ≃ Supp (Y)$ , or if $Cl_{a} (X)$ fails to agree with $Cl_{a} (Y)$ for some $a \in Σ$ , then $X \neq ≃ Y$ .

21Substrate and organization are independent

The two invariants should not be collapsed. There can exist systems $X$ and $Y$ with $Supp (X) \neq ≃ Supp (Y)$ while both are $Σ$ -closed. A hypothetical self-maintaining artificial organism could instantiate a different substrate and still satisfy the same closure axioms. Conversely, two systems built from similar materials may differ in organizational closure.

Neither sameness of material nor sameness of computation is sufficient for equivalence. The invariant that matters is structured organization.

22Simulation is weaker than equivalence

Suppose a semantics or simulation functor $S : PhysSys \to Comp$ sends a biological system $B$ to a computational representation $S (B)$ . Even a perfect reproduction of some state-transition structure is only a morphism, or an equivalence, at the semantic level. One cannot infer $B ≃ S (B)$ . The two need not even lie in the same category.

If $S$ forgets $Supp (B)$ or $Cl_{Σ} (B)$ , then $S$ fails to be conservative for precisely the structure at issue.

Categorically, this is the difference between preservation of selected observables and equivalence of structured objects.

23Open-system category theory

The framework is compatible with compositional treatments of open physical and chemical systems. Baez and Pollard model open reaction networks as morphisms, with composition given by joining outputs of one system to inputs of another. Supplementation $X ∣ H$ is a higher-categorical analogue of that composition.

The additional structure here is the separation of generic material flows from organizational enabling processes. Only the latter enter the closure invariant.

24Relation to Rosen

Rosen’s $(M, R)$ -systems motivate the same structural separation. A system may stay open to material flow while requiring that its efficient causes be generated inside the organizational network. Later analyses call this closure to efficient causation: the catalysts or processors the system requires are themselves produced within the system.

The inequality $Req_{a} (X) \leq End_{a} (X)$ is an $(\infty)$ -topos-theoretic abstraction of that relation. It does not adopt Rosen’s stronger, and controversial, claims about computability. The separation theorem uses only invariance of closure. It does not claim that organisms are noncomputable.

25Relation to autopoiesis

Maturana’s formulation requires that an autopoietic system recursively produce the components realizing its own network of production, and its boundary. The present formalism encodes this by including modes $a_{boundary}$ and $a_{production}$ and requiring both $Cl_{a_{boundary}} (X)$ and $Cl_{a_{production}} (X)$ .

A categorical formalization of autopoiesis is still an active research problem, not a settled theorem. The construction here is an axiomatic formalization, not a claim to be the unique mathematics of life.

26Main separation theorem

Theorem 26.1

Organism, virus, and computer

Let $B$ , $V$ , and $C$ in $Sys_{K, Σ} (E)$ represent a cellular organism, a virus, and a conventional computer. Assume:

Cellular closure: $Cl_{Σ_{org}} (B)$ . Viral host dependence: for some $a \in Σ_{org}$ , $\neg Cl_{a} (V)$ , but $Cl_{a} (V ∣ H)$ for a host $H$ . Computer industrial dependence: for some $b \in Σ_{org}$ , $\neg Cl_{b} (C)$ , but $Cl_{b} (C ∣ I)$ for an industrial environment $I$ .

Then $B \neq ≃ V$ and $B \neq ≃ C$ . If in addition $Supp (V) \neq ≃ Supp (C)$ , then $V \neq ≃ C$ as $K$ -typed systems, even when a dependency functor satisfies $U (V) ≃ U (C)$ .

Proof. The first two non-equivalences are Theorem 7.1: $B$ is closed in a mode where $V$ is not, and in a mode where $C$ is not. The third is Theorem 3.1. The possible equivalence $U (V) ≃ U (C)$ does not affect the conclusion, because $U$ is not conservative.

End of proof.

27Logical form of the conclusion

The invariant takes different values on the three systems:

B ↓ I I (B) V ↓ I I (V) C ↓ I I (C),

with $I (B) \neq = I (V)$ and $I (B) \neq = I (C)$ . A coarse functor may still satisfy $U (V) ≃ U (C)$ .

V \neq ≃ C

is compatible with a valid analogy $V \sim C$ in a quotient semantics. More importantly, whenever autonomous organizational closure is among the preserved structures,

B \neq ≃ V and B \neq ≃ C .

28Final categorical formulation

There is an $(\infty)$ -category $Sys_{K, Σ} (E)$ of physically typed, organizationally structured systems, and full subcategories $Aut_{Σ} (E)$ , $HostDep_{Σ} (E)$ , and $ExtMaint_{Σ} (E)$ , cut out respectively by

Cl_{Σ} (X), \neg Cl_{Σ} (X) \land \exists H . Cl_{Σ} (X ∣ H), \neg Cl_{Σ} (X) \land \exists I . Cl_{Σ} (X ∣ I) .

The second and third predicates have the same logical shape. What differs is the supplement they quantify over — host machinery, or an external maintenance ecology — together with every other invariant the ambient category retains. At a biological scale one may model $B \in Aut_{Σ}$ , $V \in HostDep_{Σ}$ , and $C \in ExtMaint_{Σ}$ . There is then no equivalence $B ≃ V$ or $B ≃ C$ preserving the full $(K, Σ)$ -structure.

A computer, a virus, and a bacterium become indistinguishable only after one successively forgets the structures that distinguish them. The mathematical question is never simply whether both can be represented computationally.

Without an affirmative answer to the last question, sameness of representation is not sameness of system. Essay II lifts the image inequality to operadic generation. Essay III requires that generation be thermodynamically feasible and that a viability kernel be inhabited.

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