Category Theory for Multi-Agent Systems¶
Overview¶
Category theory provides a mathematical foundation for modeling multi-agent systems in Timestep. This formal approach enables us to reason about agent composition, handoffs, tool usage, and workflow execution using rigorous mathematical structures.
Core Concepts¶
Categories¶
A category consists of: - Objects: The entities in the system (agents, tools, sessions, workflows) - Morphisms: The relationships/transformations between objects (handoffs, tool invocations, state transitions) - Composition: How morphisms combine (sequential handoffs, tool chaining) - Identity: Trivial morphisms (self-handoffs, no-op operations)
Category of Agents (Agt)¶
- Objects: Individual agents (e.g.,
weather_agent,assistant) - Morphisms: Handoffs between agents (
handoff: A → B) - Composition: Sequential handoffs form composition
- Identity: Self-handoff (trivial)
Category of Tools (Tool)¶
- Objects: Individual tools (e.g.,
get_weather,WebSearchTool) - Morphisms: Tool invocations (
invoke: Tool → Result) - Composition: Tool chaining (when tools output feeds into next tool)
Functors¶
Functors map between categories while preserving structure:
- Agent-Tool Functor: Maps agents to their available tools
- Handoff Functor: Maps agents to their handoff targets (delegation structure)
- State Functor: Maps agent-session pairs to execution states
Monoidal Structure¶
The monoidal structure enables parallel composition:
- Tensor product: Parallel agent execution
- Unit: Empty agent (identity)
- Composition: Sequential agent workflows
Benefits¶
- Formal Verification: Prove properties about agent compositions (associativity, identity, termination)
- Type Safety: Verify tool outputs match expected inputs
- Composition: Reason about complex agent workflows mathematically
- Optimization: Identify redundant patterns and optimize compositions
Implementation¶
See the analysis module for the implementation of these concepts in both Python and TypeScript.