Categorical Morphisms & Kleisli Composition¶

The algebrax.category module models categories where hom-sets \(\text{Hom}(A, B)\) are represented as sparse matrix mappings \(M[A][B]\).

Effectful monadic morphisms \(f: A \to T(B)\) and \(g: B \to T(C)\) compose via Kleisli Composition:

\[(g \circ_T f)(A, C) = \bigoplus_{B} f(A, B) \otimes g(B, C)\]

Semiring Monad Enrichment¶

By swapping the underlying semiring, the exact same kleisli_compose() function computes different monadic effects: - ViterbiSemiring: Probabilistic transition chains (\(f: A \to \text{Prob}(B)\)). - TropicalSemiring: Lawvere metric space minimum cost path composition (\(f: A \to \text{Cost}(B)\)). - BooleanSemiring: Poset graph reachability (\(f: A \to \text{Bool}(B)\)).

Python Example: Kleisli Monadic Composition¶

import algebrax as ax

# Define Morphisms A -> B and B -> C
f_prob = {'A': {'B': 0.8, 'C': 0.2}}
g_prob = {'B': {'D': 0.9}, 'C': {'D': 0.5}}

# 1. Probabilistic Viterbi Monad
prob_res = ax.category.kleisli_compose(f_prob, g_prob, semiring=ax.semiring.ViterbiSemiring())
print("Probabilistic Max Path A -> D:", prob_res['A']['D'])
# Output: 0.72

# 2. Lawvere Metric Cost Monad
cost_res = ax.category.kleisli_compose({'A': {'B': 3.0}}, {'B': {'D': 2.0}}, semiring=ax.semiring.TropicalSemiring())
print("Min Shortest Path Cost A -> D:", cost_res['A']['D'])
# Output: 5.0

# 3. Boolean Reachability Monad
bool_res = ax.category.kleisli_compose({'A': {'B': True}}, {'B': {'D': True}}, semiring=ax.semiring.BooleanSemiring())
print("Boolean Reachability A -> D:", bool_res['A']['D'])
# Output: True