---
title: "Categories for the Working Mathematician (2nd ed.)"
authors:
  - "Saunders Mac Lane"
year: 1998
venue: "Springer Graduate Texts in Mathematics 5"
arxiv: null
doi: "10.1007/978-1-4757-4721-8"
source: "https://link.springer.com/book/10.1007/978-1-4757-4721-8"
topics:
  - category-theory
  - foundations
  - monads
  - kan-extensions
seed_rank: 29
seed_batch: "prefill-2026-08-13"
reviewed: "2026-08-13"
pool: "maths-foundations"
relevance_score: 10
cites:
  - title: "General Theory of Natural Equivalences"
    url: "https://doi.org/10.2307/1990284"
    year: 1945
    arxiv: null
    doi: "10.2307/1990284"
  - title: "Computational lambda-calculus and monads"
    url: "https://doi.org/10.1109/LICS.1989.39155"
    year: 1989
    arxiv: null
    doi: "10.1109/LICS.1989.39155"
see:
  - "050-general-theory-of-natural-equivalences"
  - "037-computational-lambda-calculus-and-monads"
---

# Categories for the Working Mathematician (2nd ed.)

## One-sentence takeaway

Mac Lane’s GTM 5 is the standard graduate text: categories, functors, natural transformations, limits, adjoints, monads and their algebras, Kan extensions, and monoidal / 2-categorical chapters added in the 1998 second edition.

## Why it matters here

When Broadside or ano notes say “the monad is an adjunction” or “Kan extend the embedding,” this is the book they mean. Pierce is the CS on-ramp; this is the reference.

## Key ideas

- The Eilenberg–Mac Lane vocabulary (1945) is developed into a working toolkit: Yoneda, representability, limits/colimits as adjoints to the diagonal, and the adjoint functor theorems.
- Monads (triples) and their Eilenberg–Moore and Kleisli categories are treated as algebra, which is the formal source of Moggi’s later computational monads.
- Kan extensions are presented as the uniform construction that specializes to limits, adjunctions, and density — the “everything is a Kan extension” slogan starts here.
- The second edition adds more on monoidal and 2-categories, which is what Baez–Stay and later applied CT actually use.
- Springer GTM 5, DOI 10.1007/978-1-4757-4721-8; first edition 1971.

## Caveats

## Links

- DOI: [10.1007/978-1-4757-4721-8](https://doi.org/10.1007/978-1-4757-4721-8)
- Springer: https://link.springer.com/book/10.1007/978-1-4757-4721-8