A Category Theory Account of AI Identity

Abstract: AI systems are routinely modified after deployment through retraining and changes in their environments. These transformations raise a metaphysical question: under what conditions does an AI system remain the same system over time or across deployments? Earlier work formulates synchronic and diachronic identity propositionally, by relating identity within a fixed AI system type to equality of trustworthiness levels. Such criteria specify when identity statements are true, but leave implicit the structure of the states compared, the transformations connecting them, and the temporal organization of persistence. We develop a category-theoretic formalization of AI identity. An AI system type is specified by a datum consisting of a techno-function, a trustworthiness profile, and a trustworthiness-level function. Profile-relative states are connected by admissible lifecycle paths, which are restricted to trustworthiness-level-preserving transformations and quotiented to obtain a reachability category. Temporally admissible functors represent AI system histories, while time-synchronous natural transformations compare realized histories. The formalization yields two categorical interpretations of the earlier AI identity criteria. A weak interpretation recovers identity as equality of trustworthiness level. A strong interpretation requires mutual trustworthiness-preserving reachability, expressed through state isomorphism or natural isomorphism of realized histories. Category theory therefore replaces a single AI identity relation with a structured hierarchy of diachronic and synchronic criteria. The resulting framework identifies identity-related preconditions for transferring responsible-AI claims, evidence, and governance procedures across versions or deployments, without treating categorical identity as sufficient by itself for such transfer.

• The paper advances a categorical framework that formalizes AI system identity based on trustworthiness and techno-function concepts.
• The framework utilizes category-theoretic constructions of states, paths, and histories to model lifecycle transformations and identity interruptions.
• The approach provides practical criteria for regulatory accountability and evidence transfer in dynamically evolving AI systems.

A Category-Theoretic Formalization of AI System Identity

Introduction

This paper presents a categorical framework for analyzing the identity of AI systems under dynamic transformation. Recognizing that AI systems experience myriad post-deployment modifications—ranging from retraining and pipeline adjustment to environmental change—the work targets a central problem for governance: When does an evolving AI system count as "the same" system for purposes of responsibility, evidence transfer, and regulatory assessment? Existing governance-adjacent criteria for AI identity, notably those based on trustworthiness levels, are primarily propositional and do not elucidate the structural, temporal, or transformational relationships necessary for a fine-grained metaphysics or a robust framework for regulatory transfer. The paper advances the state of AI artifact metaphysics by developing a detailed, rigorous category-theoretic treatment of AI identity grounded in the techno-function and trustworthiness paradigm.

The Trustworthiness-Based Foundation of AI Identity

The formal account builds on a trustworthiness-centric metaphysics of artifacts, itself an adaptation of the function+ account established by Carrara and Vermaas. In this paradigm, an AI system's identity is anchored not solely in its abstract function or material composition, but in the conjunction of:

1. Techno-function ($F$): The high-level technical capability an AI system is engineered to realize.
2. Trustworthiness profile ($P$): A structured, finite set of governance-relevant requirements (performance, robustness, fairness, explainability, etc.), tied to explicit measurement procedures and thresholds.
3. Trustworthiness-level function ($L_p$): A mapping from quantitative assessments of $P$ to a discrete set of governance-relevant trustworthiness levels. This function adjudicates which quantitative states are considered sufficiently similar at the level required for regulatory continuity and evidential transfer.

The type datum $\langle F, P, L_p \rangle$ thus specifies both the design intent and governance envelope for AI system kinds.

Category-Theoretic Construction

States, Paths, and Thin Categories

Given a type datum, an AI state is a tuple ($q_P$, $L_p(q_P)$) specifying a particular quantitative valuation of trustworthiness dimensions and its corresponding trustworthiness level. The set of all such states, $S_\mathcal{A}$, serves as the object class for subsequent categorical constructions.

Primitive lifecycle interventions (e.g., retraining, recalibration) are formalized as labeled relations on this space. These generate a free path category $\text{Path}_\mathcal{A}$ whose morphisms are finite sequences of admissible transitions, preserving lifecycle provenance.

Critically, the path category is restricted to paths preserving trustworthiness level ($\text{Path}_\mathcal{A}^\ell$), and then quotiented by an equivalence relation identifying parallel paths with the same endpoints and trustworthiness level. The result is $P$0, a thin category (i.e., a preorder): for each pair of states, there is at most one morphism, reflecting level-preserving reachability and abstracting from concrete lifecycle provenance.

Time and System Histories

To address diachronic (temporal) identity, the framework enriches $P$1 with explicit temporality by treating time as a thin category (poset), $P$2. AI system histories are defined as functors $P$3, assigning to each time point a state and to each temporal interval a (trustworthiness-preserving) morphism. Temporal admissibility is enforced by requiring the existence of time-indexed representatives that respect the temporal sequence encountered in practice.

The paper distinguishes abstract system histories from realized histories, the latter being those functors for which there exists at least one actually deployed AI token instantiating the prescribed sequence of states over the interval.

Comparison and Identity of Histories

The comparison of realized histories is captured categorically via natural transformations between functors, with the restriction that each component (for each time $P$4) possesses a time-synchronous representative—i.e., a path staying within the set of states actually realized at $P$5. The resulting category $P$6 has realized histories as objects and time-synchronous natural transformations as morphisms.

Hierarchy of Identity Criteria

The categorical formalization provides a refined hierarchy of identity notions:

• Weak identity: Equated with coincidence of trustworthiness level (recovers previous propositional approaches).
• Directed reachability: The existence of a trustworthiness-level-preserving path from one state to another (not necessarily invertible).
• Strong identity (state-level): Isomorphism in $P$7; i.e., mutual reachability via level-preserving paths, reflecting bidirectional compatibility.
• Weak synchronic identity of histories: Existence of a natural transformation between realized histories (time-synchronous comparison at each instant).
• Strong synchronic identity of histories: Natural isomorphism in $P$8; i.e., mutual time-synchronous comparability at every time point.

Changes in trustworthiness level are shown to induce identity interruption: no morphism exists between states in different level fibers in $P$9, so a level drop fragments the identity-preserving history.

Implications for AI Governance and Artifact Metaphysics

The major implications are both theoretical and practical. From a metaphysical perspective, the categorical framework distinguishes weak and strong notions of identity, accommodates both endurantist and perdurantist views, and grounds identity in temporally indexed, governance-relevant invariants rather than in raw material or computational continuity.

Practically, it provides explicit criteria for when properties, evidence, certifications, and accountability can be transferred across system versions or instances—crucial in regulatory regimes such as the EU AI Act. The framework also clarifies the impact of changes in measurement granularity and trustworthiness-level design, highlighting the necessity for robust, auditably stable level functions to ensure identity persistence amid operational fluctuations.

Moreover, the categorical approach underlines why identity cannot be entirely provenance-sensitive: for governance, only properties invariant under admissible transformations (as abstracted in the quotient category) are identity-preserving. Conversely, provenance remains recoverable at the level of the free path category for applications where lifecycle audit trails are required.

The category-theoretic language proves essential for articulating compositionality, abstraction, and transformational relations in a unified, mathematically precise manner, enabling generalization and extension to scenarios involving interconnected or multi-agent systems.

Prospects for Future Developments

Further development could operationalize these categorical criteria within dynamic MLOps pipelines and regulatory monitoring systems, providing automated tracking of identity-preserving interventions, and more nuanced classification of substantial modifications vs. identity-preserving upgrades. The extension of the hierarchical framework to more general monoidal or higher-categorical structures may facilitate reasoning about networks of interacting AI systems, multi-agent environments, or federated governance.

Additional work is warranted to develop epistemologies and toolchains for recognizing and certifying identity relations in practice, leveraging documentation and automated logging requirements already mandated by emerging AI regulation.

Conclusion

This work establishes a formally robust, governance-focused categorical model of AI system identity, clarifying the preconditions for sameness across lifecycle changes. By distinguishing weak and strong identity paradigms, anchoring identity in techno-function and trustworthiness, introducing transformation-based, temporally indexed structure, and elucidating the circumstances under which identity is interrupted, the framework creates the foundations for both philosophical analysis and practical governance of AI system persistence. Application of the framework in operational and regulatory contexts will depend on the careful design of trustworthiness profiles, level functions, and supporting documentation to ensure stability and traceability of identity across the AI lifecycle.