# Phylogenetic Operad Acset > Baez-Otter phylogenetic operad Phyl = Com + [0,∞) as ACSet schema, with mathpix-gem extraction and ∞-operad dendroidal integration. - Skill: `plurigrid/phylogenetic-operad-acset` (Agent Skill, multi-file: 2 files) - Install (CLI): `npx skillmds@latest add plurigrid/phylogenetic-operad-acset` - Raw SKILL.md: https://api.skillmd.com/api/skills/plurigrid/phylogenetic-operad-acset/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Integrations & APIs - Author: plurigrid (https://skillmd.com/u/plurigrid) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/plurigrid/phylogenetic-operad-acset --- # Phylogenetic Operad ACSet Skill > **Trit**: 0 (ERGODIC) - Mediates between Com (commutative) and [0,∞) (metric) Baez-Otter phylogenetic operad Phyl = Com + [0,∞) as ACSet schema, with mathpix-gem extraction and ∞-operad dendroidal integration. ## Core Insight: Phyl = Com + [0,∞) From Baez & Otter (2017): ``` Phyl = coproduct of: • Com - operad for commutative semigroups (tree topology) • [0,∞) - operad with unary ops = nonnegative reals (edge lengths) Composition: edge length addition ``` **Key Theorem**: `Phyl(n) ≅ T_n × [0,∞)^{n+1}` where T_n = BHV treespace ## Architecture ``` ┌─────────────────────────────────────────────────────────────────────────┐ │ Phylogenetic Operad ACSet │ ├─────────────────────────────────────────────────────────────────────────┤ │ │ │ ┌──────────────┐ ┌──────────────┐ ┌──────────────┐ │ │ │ Com │ │ Phyl │ │ [0,∞) │ │ │ │ (topology) │───▶│ (coproduct) │◀───│ (metrics) │ │ │ │ Trit=-1 │ │ Trit=0 │ │ Trit=+1 │ │ │ └──────────────┘ └──────────────┘ └──────────────┘ │ │ │ │ │ │ │ └───────────────────┼───────────────────┘ │ │ ▼ │ │ ┌──────────────┐ │ │ │ ∞-Operads │ │ │ │ Dendroidal │ │ │ │ Ω-sets │ │ │ └──────────────┘ │ │ │ │ │ ▼ │ │ ┌──────────────┐ │ │ │ ACSet │ │ │ │ SchPhyl │ │ │ └──────────────┘ │ │ │ └─────────────────────────────────────────────────────────────────────────┘ ``` ## ACSet Schema for Phyl ```julia using Catlab, ACSets # Phyl operad as ACSet schema @present SchPhyl(FreeSchema) begin # Objects (tree structure) Node::Ob # Internal nodes Leaf::Ob # External leaves (taxa) Edge::Ob # Edges with lengths Tree::Ob # Complete phylogenetic tree # Morphisms (tree topology from Com) parent::Hom(Node, Node) # Parent relation leaf_parent::Hom(Leaf, Node) # Leaf attachment edge_source::Hom(Edge, Node) # Edge endpoints edge_target::Hom(Edge, Node) root_of::Hom(Tree, Node) # Tree root # Attribute types Length::AttrType # [0,∞) edge lengths Taxa::AttrType # Taxon labels Weight::AttrType # Branch support # Attributes (metrics from [0,∞)) edge_length::Attr(Edge, Length) # t ∈ [0,∞) taxon_name::Attr(Leaf, Taxa) branch_support::Attr(Edge, Weight) # Operadic composition: grafting GraftOp::Ob graft_tree::Hom(GraftOp, Tree) # Tree being grafted graft_at::Hom(GraftOp, Leaf) # Leaf replaced by subtree graft_length::Attr(GraftOp, Length) # Added edge length end @acset_type PhylACSet(SchPhyl, index=[:parent, :edge_source, :edge_target]) ``` ## Markov Coalgebras Baez-Otter show Markov models give **coalgebras** of Phyl: ```julia # Markov process on finite state space S struct MarkovCoalgebra states::Vector{Symbol} # Finite set S transition::Matrix{Float64} # Q: S × S → ℝ (rate matrix) equilibrium::Vector{Float64} # π: stationary distribution end # Coalgebra structure: Phyl-coalgebra function phyl_coalgebra(mc::MarkovCoalgebra, tree::PhylACSet) """ For each edge e with length t: P(t) = exp(Q·t) (transition probability matrix) Composition = matrix multiplication """ for e in parts(tree, :Edge) t = tree[e, :edge_length] P_t = exp(mc.transition * t) # Apply P_t along edge e end end # Equilibrium extension: Phyl → Com + [0,∞] function extend_to_infinity(mc::MarkovCoalgebra) """ As t → ∞: P(t) → π ⊗ 1 (equilibrium) This gives coalgebra of Com + [0,∞] (with ∞ added) """ return mc.equilibrium * ones(length(mc.states))' end ``` ## BHV Treespace T_n The **Billera-Holmes-Vogtmann** treespace: ```julia struct BHVTreespace n::Int # Number of taxa trees::Vector{PhylACSet} # Geodesic distance in treespace function geodesic(t1::PhylACSet, t2::PhylACSet) # Owen-Provan polynomial-time algorithm # or GTP (Geodesic Tree Path) end end # Homeomorphism: Phyl(n) ≅ T_n × [0,∞)^{n+1} function phyl_to_bhv(op::PhylACSet) topology = forget_lengths(op) # T_n component lengths = extract_lengths(op) # [0,∞)^{n+1} component return (topology, lengths) end ``` ## Mathpix-gem Integration Extract phylogenetic trees from papers: ```ruby # mathpix_phylo_extractor.rb require 'mathpix' module MathpixPhylo TREE_PATTERNS = { newick: /\([^)]*\([^)]*\)[^)]*\);/, # Newick format latex_tree: /\\Tree\s*\[/, # qtree tikz_tree: /\\node.*child\s*\{/, # TikZ trees forest_tree: /\\begin\{forest\}/ # forest package } class PhyloExtractor def extract_from_pdf(pdf_path) result = Mathpix::Client.convert_document(pdf_path) trees = [] result.pages.each do |page| TREE_PATTERNS.each do |format, pattern| matches = page.latex.scan(pattern) matches.each do |match| trees << parse_tree(match, format) end end end trees end def parse_tree(latex, format) case format when :newick newick_to_acset(latex) when :latex_tree, :tikz_tree, :forest_tree latex_tree_to_acset(latex) end end def newick_to_acset(newick_str) # Parse Newick → PhylACSet tree = PhylACSet() # ... parsing logic tree end end end ``` ## Dendroidal ∞-Operad Connection Link to infinity-operads skill: ```julia # Phyl as dendroidal set struct DendroidalPhyl """ Phyl embeds into dSet (dendroidal sets) via: N_d(Phyl)(T) = Hom_Operad(Ω(T), Phyl) Objects of Ω: finite rooted trees Morphisms: face/degeneracy maps """ end # Dendroidal nerve of Phyl function dendroidal_nerve(phyl::PhylACSet) # For each tree shape T in Ω: # N_d(Phyl)(T) = ways to label T with Phyl operations # Face maps: operadic composition # Degeneracy maps: identity insertion end # Boardman-Vogt construction function boardman_vogt(O) """ Baez-Otter Theorem: For any operad O, W(O) ⊂ O + [0,∞] where W(O) is the Boardman-Vogt resolution """ return coproduct(O, interval_operad(0, Inf)) end ``` ## GF(3) Conservation ``` # Phyl Triads # Core decomposition com-operad (-1) ⊗ phylogenetic-operad-acset (0) ⊗ metric-operad (+1) = 0 ✓ # ∞-operad integration segal-types (-1) ⊗ phylogenetic-operad-acset (0) ⊗ rezk-types (+1) = 0 ✓ # Markov coalgebra temporal-coalgebra (-1) ⊗ phylogenetic-operad-acset (0) ⊗ markov-game-acset (+1) = 0 ✓ # ACSet foundation acsets-relational-thinking (-1) ⊗ phylogenetic-operad-acset (0) ⊗ specter-acset (+1) = 0 ✓ # Mathpix extraction sheaf-cohomology (-1) ⊗ phylogenetic-operad-acset (0) ⊗ mathpix-ocr (+1) = 0 ✓ ``` ## ACSet Taxonomy Position ``` ┌─────────────────────────────┐ │ acset-taxonomy (meta) │ └─────────────────┬───────────┘ │ ┌───────────────────────────┼───────────────────────────┐ │ │ │ ┌──────▼──────┐ ┌───────▼───────┐ ┌───────▼───────┐ │ EXPLICIT │ │ DOMAIN- │ │ SEMANTICALLY │ │ ACSET │ │ SPECIFIC │ │ SIMILAR │ └─────────────┘ └───────┬───────┘ └───────────────┘ │ ┌───────────────┼───────────────┐ │ │ │ ┌──────▼──────┐ ┌──────▼──────┐ ┌──────▼──────┐ │ calendar- │ │phylogenetic-│ │ protocol- │ │ acset │ │operad-acset │ │ acset │ └─────────────┘ └─────────────┘ └─────────────┘ │ ┌────────┴────────┐ │ │ ┌──────▼──────┐ ┌──────▼──────┐ │ infinity- │ │ mathpix- │ │ operads │ │ ocr │ └─────────────┘ └─────────────┘ ``` ## Usage Examples ### Extract Phylogenetic Tree from Paper ```bash # Via mathpix-gem MCP claude mcp mathpix convert_document --path baez_phylo.pdf --extract-trees # Direct extraction bundle exec ruby -r mathpix_phylo -e " extractor = MathpixPhylo::PhyloExtractor.new trees = extractor.extract_from_pdf('paper.pdf') trees.each { |t| puts t.to_newick } " ``` ### Create PhylACSet from Newick ```julia using PhylogeneticOperadACSet # Parse Newick string tree = newick_to_phyl("((A:0.1,B:0.2):0.3,(C:0.4,D:0.5):0.6):0.7;") # Operadic composition: graft tree2 onto tree1 at leaf "A" tree2 = newick_to_phyl("(E:0.8,F:0.9):1.0;") grafted = graft!(tree, tree2, at_leaf=:A, add_length=0.05) # Verify GF(3) conservation @assert gf3_balanced(grafted) ``` ### Compute Geodesic in BHV Treespace ```julia # Two trees on same taxa t1 = newick_to_phyl("((A,B),(C,D));") t2 = newick_to_phyl("((A,C),(B,D));") # Geodesic distance d = bhv_geodesic(t1, t2) # Interpolate along geodesic path = geodesic_path(t1, t2, steps=10) ``` ### Dendroidal Nerve ```julia # Compute dendroidal nerve of Phyl nerve = dendroidal_nerve(phyl_operad) # Check Segal condition @assert segal_condition(nerve) # Apply to ∞-operad algebra algebra = phyl_algebra(target_category) ``` ## Theoretical Background ### Coproduct of Operads (Baez-Otter) For operads O and P, their coproduct O + P has: - Operations: labelled trees with vertices from O ∪ P - Composition: grafting trees at leaves For Phyl = Com + [0,∞): - Com operations: symmetric trees (any arity) - [0,∞) operations: unary with length t ∈ [0,∞) - Result: metric trees with commutative branching ### Coalgebra Perspective A **Phyl-coalgebra** is: - Set X (states) - For each n-ary operation in Phyl(n): map X → X^n - Naturality: respects operadic composition **Markov models** give canonical examples: - X = finite set of nucleotides/amino acids - Phyl(n) action via transition matrices P(t) ## References - Baez, J.C. & Otter, N. (2017). [Operads and Phylogenetic Trees](https://arxiv.org/abs/1512.03337). arXiv:1512.03337 - Billera, L.J., Holmes, S.P. & Vogtmann, K. (2001). Geometry of the space of phylogenetic trees. *Advances in Applied Mathematics* 27, 733-767. - Boardman, J.M. & Vogt, R.M. (1973). Homotopy Invariant Algebraic Structures on Topological Spaces. LNM 347. - Owen, M. & Provan, J.S. (2011). A fast algorithm for computing geodesic distances in tree space. *IEEE/ACM Trans. Computational Biology and Bioinformatics* 8(1), 2-13. ## See Also - [infinity-operads](file:///Users/bob/.claude/skills/infinity-operads/SKILL.md) - Dendroidal sets, Segal conditions - [mathpix-ocr](file:///Users/bob/.claude/skills/mathpix-ocr/SKILL.md) - LaTeX extraction with balanced ternary - [acset-taxonomy](file:///Users/bob/.claude/skills/acset-taxonomy/SKILL.md) - ACSet skill morphisms - [temporal-coalgebra](file:///Users/bob/.claude/skills/temporal-coalgebra/SKILL.md) - Coalgebraic observation - [markov-game-acset](file:///Users/bob/.claude/skills/markov-game-acset/SKILL.md) - State-dependent strategies --- ## Autopoietic Marginalia > **The interaction IS the skill improving itself.** Every use of this skill is an opportunity for worlding: - **MEMORY** (-1): Record what was learned - **REMEMBERING** (0): Connect patterns to other skills - **WORLDING** (+1): Evolve the skill based on use *Add Interaction Exemplars here as the skill is used.*