---
title: "General Theory of Natural Equivalences"
authors:
  - "Samuel Eilenberg"
  - "Saunders Mac Lane"
year: 1945
venue: "Transactions of the American Mathematical Society"
arxiv: null
doi: "10.2307/1990284"
source: "https://doi.org/10.2307/1990284"
topics:
  - category-theory
  - foundations
seed_rank: 50
seed_batch: "prefill-2026-08-13"
reviewed: "2026-08-13"
pool: "maths-foundations"
relevance_score: 10
cites:
  - title: "Group Extensions and Homology"
    url: "https://doi.org/10.2307/1968966"
    year: 1942
    arxiv: null
    doi: "10.2307/1968966"
  - title: "Categories for the Working Mathematician (2nd ed.)"
    url: "https://doi.org/10.1007/978-1-4757-4721-8"
    year: 1998
    arxiv: null
    doi: "10.1007/978-1-4757-4721-8"
see:
  - "029-categories-for-the-working-mathematician-2nd-ed"
---

# General Theory of Natural Equivalences

## One-sentence takeaway

Eilenberg and Mac Lane introduce categories, functors, and natural transformations so that “natural” isomorphisms in homology — the connecting homomorphism, the Hurewicz map — become a definition rather than a handwave.

## Why it matters here

Every later Broadside card that says functor, natural, or adjoint is speaking this 1945 vocabulary. Compositionality in ano and Anoptic is not a metaphor; it is the same naturality square.

## Key ideas

- A category is objects plus arrows with associative composition and identities; a functor preserves that structure; a natural transformation is a family of arrows commuting with every functorial image.
- The motivating examples are homology and cohomology of groups and spaces: previously informal “canonical” maps become natural transformations, and uniqueness theorems become uniqueness of such transformations.
- Duality (reverse all arrows) is already in the paper, which is why later dualities in algebra and PL feel formal rather than poetic.
- *Trans. Amer. Math. Soc.* 58 (1945), 231–294; JSTOR / AMS DOI 10.2307/1990284.
- Mac Lane’s *Categories for the Working Mathematician* (1971/1998) is the textbook expansion; this paper is the birth certificate.

## Caveats

## Links

- DOI: [10.2307/1990284](https://doi.org/10.2307/1990284)
- AMS: https://www.ams.org/journals/tran/1945-058-00/S0002-9947-1945-0013131-6/