Core Concepts

Diagrams

A Diagram is the package's central representation of a structured AI system. Objects define interfaces, morphisms define transformations, Kan extensions define universal aggregation or completion, and obstruction losses encode consistency constraints.

Kan extensions

FunctorFlow uses categorical language directly:

Σ / left Kan extension for aggregation, pooling, attention, and message passing
Δ / right Kan extension for completion, repair, conditioning, and reconciliation

These are available both as functions and as Unicode operators.

Obstruction losses

An ObstructionLoss measures how far a diagram is from commuting. This is the package's basic mechanism for diagrammatic consistency:

symbolic workflows can compare strings, sets, or structured values
neural workflows can optimize differentiable comparators such as :l2, :l1, or cosine-style losses

Compilation pipeline

FunctorFlow separates specification from execution:

Diagram
  -> DiagramIR
  -> CompiledDiagram
  -> callable or Lux-backed executable model

The same categorical structure can therefore be used for both symbolic and differentiable programs.

Beyond the core

The package also includes higher-level categorical layers:

universal constructions such as pullback, pushout, product, and coequalizer
causal semantics via CausalContext and build_causal_diagram
topos-style reasoning tools such as SubobjectClassifier
coalgebra and JEPA-inspired world-model structures

These features extend FunctorFlow from a DSL for blocks into a broader neurosymbolic research toolkit.