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LIT-803: Seven Sketches in Compositionality: An Invitation to Applied Category Theory

3 min read

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StatusActive — skimmed 2026-10-09 (NOTE-601): the preface and Chapter 1 read closely, Chapter 3's opening and its section on adjunctions and data migration read closely, the openings and summaries of Chapters 2 and 4–7 read; Chapters 2 and 4–7 otherwise not read. Worth reading as an introduction to applied category theory organised around compositionality: an observation of systems is a monotone map, a generative effect is its failure to preserve joins, and a monotone map out of a preorder with all joins has no generative effect exactly when it is a left adjoint. Later chapters treat resource theories, databases as functors, co-design, signal-flow graphs, circuits and temporal logic, each through a categorical structure.
Tagsmathematics, compositionality
Cited as “paper” byNOTE-601 — Seven Sketches in Compositionality
Cited as “source” byTHEORY-686 — An observation modelled as a monotone map between preorders can show of a joined system more than the join of its observations of the parts, and out of a preorder with all joins the observations that preserve all joins, the empty join and infinite joins included, are exactly the left adjoints

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Brendan Fong, David I. Spivak (arXiv 2018; Cambridge University Press 2019, as An Invitation to Applied Category Theory: Seven Sketches in Compositionality) — ARXIV-1803.05316, DOI-10.1017/9781108668804

Key takeaways

From the parts read (see NOTE-601 for exactly which):

  • Observation as a monotone map, and generative effects. Following Adam’s thesis, an observation of a system is a monotone map Φ : P → Q between preorders. It always satisfies Φ(a) ∨ Φ(b) ≤ Φ(a ∨ b); a generative effect is a case where the inequality is strict: observing the joined system shows something the joined observations do not. The worked example is connectivity of partitions of a three-point set, observed by “is • connected to ∗?”.
  • Left adjoints are exactly the maps without generative effects. Left adjoints preserve all joins and right adjoints all meets (Proposition 1.111); if P has all joins, a monotone map out of P preserves joins iff it is a left adjoint (Theorem 1.115). Adam’s generative observations are those that preserve meets but not joins.
  • Galois connections, closure and level shifting. f ⊣ g iff p ≤ g(f(p)) and f(g(q)) ≤ q; g∘f is a closure operator, and every closure operator comes from an adjunction with its fixed points. Each function g : S → T induces an adjunction between partitions, and each f : A → B the triple f! ⊣ f* ⊣ f∗ on subsets (image, preimage, “all in”).
  • Databases as functors. A schema is a category, an instance a functor to Set, and a schema map F induces the migrations Δ_F with adjoints Σ_F ⊣ Δ_F ⊣ Π_F (union-like and join-like); for F : C → 1, Σ gives connected components and Π a selection.
  • The book’s stance on coherence. The authors call the choice of coherence conditions “more science than mathematics”: accepted when they yield widely applicable, strongly compositional structures.

Standing in the record

Filed on 2026-10-09 at the owner’s request, from the manuscript bibliography of 2026-10-09 (work what-survives-translation): one of the works the manuscript considered and dropped from its final reference list.

Skimmed on 2026-10-09 (NOTE-601): the chapters closest to its opening aim, orders and adjunctions as the basis of compositional modelling, were read closely, and the rest only at their openings and summaries. It is an introduction, not a source for results, which are standard; a deeper reading would take Chapters 2 and 4 (resource theories and monoidal categories) and Chapter 7 (sheaves and toposes). Its Galois-connection material is the order theory underlying Ganter and Wille’s concept lattices (LIT-344), which the book does not mention.

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