---
title: "Seven Sketches in Compositionality: An Invitation to Applied Category Theory"
authors:
  - "Brendan Fong"
  - "David I. Spivak"
year: 2019
venue: "Cambridge University Press"
arxiv: "1803.05316"
doi: "10.1017/9781108668804"
source: "https://arxiv.org/abs/1803.05316"
topics:
  - category-theory
  - applied-ct
  - monoidal
  - databases
seed_rank: 94
seed_batch: "prefill-2026-08-13"
reviewed: "2026-08-13"
pool: "maths-foundations"
relevance_score: 9
cites:
  - title: "General Theory of Natural Equivalences"
    url: "https://doi.org/10.2307/1990284"
    year: 1945
    arxiv: null
    doi: "10.2307/1990284"
  - title: "Physics, Topology, Logic and Computation: A Rosetta Stone"
    url: "https://arxiv.org/abs/0903.0340"
    year: 2009
    arxiv: "0903.0340"
    doi: "10.1007/978-3-642-12821-9_2"
see:
  - "050-general-theory-of-natural-equivalences"
  - "018-physics-topology-logic-and-computation-a-rosetta-stone"
---

# Seven Sketches in Compositionality: An Invitation to Applied Category Theory

## One-sentence takeaway

Seven paired tours — each an application (databases, circuits, dynamical systems, …) next to a structure (adjuncts, enrichment, toposes, …) — invite builders into applied category theory with no CT prerequisite.

## Why it matters here

Anoptic interconnect, typed wiring, and database-shaped state are exactly the sketches this book teaches: orders, monoidal wiring diagrams, and lenses rather than free-floating abstraction.

## Key ideas

- The book is a tour, not a treatise: each sketch is meant to unlock later reading rather than exhaust a topic.
- Concrete running examples include databases, electric circuits, and dynamical systems.
- Matching structures include adjoint functors, enriched categories, monoidal categories, and toposes.
- Preprint is 341+xii pages (arXiv 1803.05316); the Cambridge volume is *An Invitation to Applied Category Theory* (DOI 10.1017/9781108668804).
- No prior category theory is assumed.

## Caveats

## Links

- arXiv: [1803.05316](https://arxiv.org/abs/1803.05316)
- PDF: https://arxiv.org/pdf/1803.05316
- DOI: [10.1017/9781108668804](https://doi.org/10.1017/9781108668804)