Conceptual Guides

This section provides an introduction to the mathematical and computational foundations of quivers. The guides assume familiarity with basic category theory or a willingness to learn it alongside the material.

  1. Core Types & Quantales: Start here. Introduces finite sets (SetObject hierarchy), quantales as enrichment algebras, and the algebraic primitives that underpin all morphism composition.

  2. Morphisms & Composition: Defines what a morphism is as a tensor in \(\mathcal{V}^{|A| \times |B|}\). Covers the morphism hierarchy, composition, and algebraic operations.

  3. Categorical Structures: Higher-order constructions: functors, natural transformations, adjunctions, monoidal structures, and base change.

  4. Monads & Comonads: Monadic abstractions (unit, multiply, map), Kleisli categories, algebras, and coalgebras.

  5. Enriched Category Theory: Advanced structures specific to \(\mathcal{V}\)-enrichment: ends, coends, Kan extensions, weighted limits, profunctors, Yoneda, Day convolution, optics.

  6. Stochastic Morphisms: The FinStoch category: Markov kernels, conditioned distributions, queries, and the Giry monad.

  7. Continuous Distributions: ContinuousSpace and ContinuousMorphism: parameterized families, sampled composition, normalizing flows.

  8. Monadic Programs: Probabilistic programming via sequential draw and let steps, ancestral sampling, and log-joint computation.

  9. The QVR DSL: Declarative specification of categorical networks: the .qvr file format, grammar, and compilation pipeline.

  10. Variational Inference: Inference pipeline: tracing, conditioning, automatic guides, ELBO, SVI, and predictive sampling.

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