ACT 2026: Applied Category Theory

ACT is the annual anchor for compositional methods across quantum theory, probability, dynamics, type systems, and applied AI. Three threads on this year's programme land directly on active wal.sh research:

1. Polynomial functors and lenses — Braithwaite/Hedges/Mihejevs bring Polylang (a working compiler for the category Poly) and its substructural type theory companion. This is the same lineage as the reversible-pipeline transforms note and the type-systems-one-tool-four-ways note.
2. Categorical probability and belief updating — six papers on Markov categories, possibilistic belief, Bayesian filtering, and sequential Monte Carlo. Direct alignment with the quantitative-economics reading and the agentic-2026 "good regulator" thread.
3. Compositional verification for control — Myers/Capucci on Lyapunov assume-guarantee, Moeller/Ames on hybrid systems as coalgebras and on categorical control-barrier / control-Lyapunov functions. Direct analogue of the goldberry frontend-invariant catalog work.

Substantial author overlap with LICS 2026 (Heunen, Kaarsgaard, Lemonnier, Hefford, Capucci) makes ACT 2026 + LICS 2026 a paired reading.

• Wang. "Definable Markov Categories" [15] [pdf] — o-minimal-definable Markov categories, positive and causal subcategory of BorelStoch.

• Myers, Capucci. "Compositionality of Lyapunov functions via assume-guarantee reasoning" [62] [pdf] — systems as lenses, (L)ISS Lyapunov functions, 2-functorial construction from tangencies.

• Cockett, Kumar, Srinivasan. "Unitary, inner product, and dagger categories" [97] [pdf] — inner-product categories as an alternate characterisation of dagger categories.
• Voorneveld, Di Giorgio, Sobocinski. "Information Leakage in Resource Theories" [72] [pdf] (arxiv:2602.04425) — abelian framed bicategories; Mayer-Vietoris / Künneth for directed structures.

• Fu, Kishida. "All Hail Kleisli: An Enriched Yoneda Embedding of Indexed Modalities into Indexed Kleisli Categories" [60] [pdf]
• Brown. "A Category-theoretic Reconstruction of Logical Expressivism" [58] [pdf] — analytic-pragmatism logical expressivism as adjunction-driven connectives.
• Wilson (P.). "Metacat: a categorical framework for formal systems" [78] [pdf] — inference rules as spans in a cartesian PROP; open-source proof-checker.

• Ortiz-Muñoz. "A Rigid Category of DNA Secondary Structures" [37] [pdf] — Watson-Crick base pairing as evaluation/coevaluation in a strict pivotal monoidal category; DisCoCat-compatible functor.
• Niu, Osgood, Srinivasan, Zelko. "Temporal sheaf theory for reconciling temporal complexity within public health modeling" [51] [pdf] — Catlab.jl implementation.
• Chamoun. "Combinatorial manifolds and Kleene's theorem, homotopically" [27] [pdf] — coreflective subcategories of relational presheaves; euclidean precubical sets.
• Ghani, Nordvall Forsberg, Fish. "Snoc Trees" [13] [pdf] — container-theoretic generalisation of snoc-lists; fold operator; sampling from inductive distributions.
• Maruyama, Nasu. "Measuring Univalence and Its Failure in Homotopy Type Theory via Synthetic Cohomology" [76] [pdf] — defect-space cohomology detects triviality in truncated defect spaces.

• Di Lavore, Román, Széles. "The Magmoid of Normalized Stochastic Kernels" [28] [pdf] — non-associative composition; front-door and back-door criteria as axioms.
• Baltieri, Virgo. "Bayesian updates from determinisation" [54] [pdf] — unifilarisation as an instance of coalgebraic determinisation; Bayesian filtering as coalgebraic transition.
• Virgo, Capucci, Baltieri, Biehl. "The universal property of possibilistic belief updating" [70] [pdf] — PY as a power object; behavioural doctrine over forward-closed predicates.
• Furter. "Sequential Monte Carlo in String Diagrams" [59] [pdf] — string diagrams of s-finite kernels → weighted particle sampling; Julia + PyTorch PPL implementations.
• Sergeant-Perthuis, Smithe. "On the Functoriality of Belief Propagation Algorithms on Finite Partially Ordered Sets" [90] [pdf] — first functoriality result for loopy belief propagation.
• Kharoof, Okay. "Possibilistic empirical models and simplicial distributions" [33] [pdf] — vertex criterion for extremality of empirical models.

• Braithwaite, Hedges, Mihejevs. "Polylang: Programming with polynomial functors and lenses" [85] [pdf] — compiler for the intrinsic simple type theory of Poly; cartesian + tensor + coproduct + composition + fixpoints.
• Braithwaite, Hedges, Mihejevs. "Substructural Type Theories Modelled by Polynomial Functors" [91] [pdf] — graded linear theory using dependent Dirichlet products; "coerasure" as comonadic modality.
• Aberlé. "Compositional Program Verification with Polynomial Functors in Dependent Type Theory" [11] [pdf] — polynomials as program interfaces, Kleisli morphisms as implementations, dependent polynomials as pre/postconditions. Formalised in Agda.

• Heunen, Kaarsgaard, Lemonnier. "One rig to control them all" [1] [pdf] — eight equations for computational control; free rig category on a base prop. Also on the LICS 2026 programme.
• Mestoudjian, Wilson (M.), Vanrietvelde, Arrighi. "Picturing general quantum subsystems" [8] [pdf]
• Wilson (M.), Hefford, Hoffreumon. "Supermaps on generalised theories" [9] [pdf] — Yoneda for categorical supermaps; stable definition of higher-order real quantum theory.
• Wilson (M.). "Higher-order circuits" [6] [pdf] — higher-order circuit theories via enrichment and cotensors in symmetric polycategories.
• Okay, Stern, Haderi, Ipek. "Double categories for adaptive quantum computation" [34] [pdf] (arxiv:2510.25915) — double port graphs; simplicial instruments; contextual fraction.
• Hefford. "Nuclearity and Trace in Monoidal Bicategories with Application to Extended CFTs" [45] [pdf]
• Comfort, de Felice. "The Delayed Stabilizer ZX-Calculus" — (PDF link missing on accepted page)
• Li, Zamdzhiev. "Quantum Coherence Spaces Revisited" [74] [pdf] (arxiv:2601.15832) — MALL model via Heisenberg-Schrödinger duality; FoSSaCS 2026.

• Stoltz. "ZX Sketch: A Browser-Based ZX-Calculus Diagram Editor" [56] [pdf] (zxsketch.com) — PyZX via Pyodide + WebAssembly; 19 rewrites, 18 simplification strategies; TikZ/SVG/PNG export.
• Voorneveld. "Box of Strings: A String Diagram Rewriting Tool" [46] [pdf]
• Wells. "Shared logic interface" [39] [pdf] — Rust + datalog via string-diagram desktop app.
• Gavranović. "TensorType: Implementing and extending Deep Learning with Types" [61] [pdf] — tensor operations checked at compile time in Idris 2; branching/recursive tensors beyond rectangular shapes.

• Moeller, Ames. "Hybrid systems as coalgebras" [87] [pdf] — Lyapunov functions as F-coalgebra morphisms into a stable target σ; recovers a broad class of stability results.
• Moeller, Ames. "A categorical approach to control Lyapunov and control barrier functions" [88] [pdf] — categorical Nagumo theorem; CLFs and CBFs in the setting of control coalgebras.
• Wilson (M.). "Agent policies from higher-order causal functions" [5] [pdf] — deterministic-POMDP policies as one-input process functions; strict separation between definite-ordered and general process functions on dec-POMDPs.
• Wang (P.). "Separation and Gluing of Explanations on Sites of Dynamical Systems" [14] [pdf] — Grothendieck site of Mealy machines over o-minimal-definable sets; gluing failure for explanations.

• Lorenz, Tull. "Causal and Compositional Abstraction" [26] [pdf] (arxiv:2602.16612) — abstractions as natural transformations; unifies constructive causal abstraction, Q-τ consistency, and interchange-intervention abstractions.
• Hefford, Wilson (M.). "BV-Categories of Spacetime Interventions" [67] [pdf] — Chu construction from duoidal categories to BV-categories; canonical spacetime model.

• Arkor, Procházková. "Representability theory for multiactegories" [47] [pdf] — unified representability across (skew) actegories, skew monoidal categories, enriched categories.
• Díaz-Caro, Ivnisky, Malherbe. "Syntactic linearity and Linear hyperdoctrines for second-order intuitionistic linear logic" [49] [pdf]

• Schellhorn. "NeSyCat Theory: Semantics in Kleisli Categories for Neuro-Symbolic AI in HaskTorch" [41] [pdf] — explicit bridge between syntax-semantic duality of categorical logic and the Haskell type system.
• Marom, Zardini, Buehler. "Compositional Verification for Nature-Derived Material Design" [96] [pdf] — Nat/Art/Spec categories; multiscale hygromorphic pinecone case study.
• Leem, Bauer, Osgood. "Multilayer Category-Theoretic Knowledge Representation of Regulatory Documents" [65] [pdf] — three-layer olog-based model for building codes.
• Llanos, Angarita, Leal, De Paiva. "Knowledge flow in math communities: a bibliometric analysis of Category Theory" — (no PDF link) 46,962 publications from 13,392 researchers via zbMath; ACT as a distinct subarea.

• Ortiz-Muñoz (proceedings above [37]).
• Nester, Voorneveld. "A Simple Categorical Calculus of Interacting Processes" [21] [pdf] — confluent + terminating rewrite system; functor into free cornering as denotational semantics.
• Nester, Kuzmin, Reimaa, Speight. "Combinatory Completeness in Structured Multicategories" [22] [pdf] — generalised combinatory completeness via faithful cartesian clubs.
• Loregian, López Díaz, Maley. "The Rosen fibration" [24] [pdf]
• Osmond. "Monoidal structures for hypergraphs and transpositionality" [35] [pdf] — funny + straight tensor products; transposition law as analogy-calculus primitive.
• Altenmüller, Duncan. "Compositional Graph Pattern Matching" [50] [pdf] — graphs form a multicategory under substitution; pattern-matching for adhesive categories.
• Comfort, de Felice. "Finite Observations, Infinite Behaviour: categorical semantics for stateful processes" — (PDF link missing on accepted page) compactness theorem for closed relations between compact Hausdorff spaces.
• Lobski, Zanasi. "Layered Monoidal Theories" [57] [pdf] — variable bit width, impedance boxes, scalable ZX and symplectic algebra.
• Jacobs, Johnson, Buckland. "Counting Votes with Multisets" [66] [pdf] — instant-runoff, De Borda, single transferable vote via commutative-monoid + functor + monad structure.
• Albert, Dubut, Goubault. "Homological Algebra in Abelian Framed Bicategories" [71] [pdf]
• Bumpus, Azevedo, Capucci, Fairbanks, Rosiak. "Algorithmic and Extremal Obstructions Through the Language of Cohomology" [77] [pdf] — presheaf Čech cohomology on VertexCover, CycleCover, OddCycleTransversal; König's theorem cohomologically.
• Earnshaw, Nester, Román. "Monoidal categories graded by partial commutative monoids" [84] [pdf] — effectful categories as a special case of PCM-graded monoidal categories; recovers Freyd categories.