Categories for the Working Mathematician

Mac Lane, S. (1971). Categories for the Working Mathematician. Springer.

Type

Book

Intellectual base

Review or monograph

Year

1971

DOI

10.1007/978-1-4612-9839-7

ISBN

9780387900360

DOI registrant

Springer Nature

Link

https://doi.org/10.1007/978-1-4757-4721-8

Cited from

computer_science, mathematics

Cited by

9 citations across 9 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Equivalence Relation
  • An equivalence relation is not the same as an isomorphism, a meta-level link Mac Lane (1971) formalises in showing that "is isomorphic to" itself constitutes an equivalence relation on a class of structured objects.

    This sourceFoundational category-theory text; treats 'is isomorphic to' as itself an equivalence relation on a class of structured objects, with isomorphism-classes as its equivalence classes.

• Higher Order Function
  • The first is composition algebra over rules: compose two rules, identify the identity rule, find an inverse rule, apply distributive laws — the same algebraic apparatus that applies to numbers now applies to rules, which is the conceptual core of morphisms-as-objects in category theory and of the proofs-as-programs correspondence.

    This sourceFoundational treatment of morphisms as first-class objects and composition algebra — the conceptual core of rules-as-objects in category theory.

    Supported in partVerified against the publisher's abstract

    The book's own abstract supports category-theoretic foundations in which morphisms and their composition are the objects of study, but it says nothing about an algebra of rules or the proofs-as-programs correspondence.

    “Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality.”

    Read by an automated reader; not a human review. How support was checked

• Injectivity
  • Equivalently, from the output you can recover which input produced it: the mapping has a left inverse on its image.

    This sourceGraduate Texts in Mathematics 5. New York: Springer-Verlag, 1971. Standard reference for monomorphisms (the categorical generalization of injective maps), left inverses, and injective objects/resolutions.

• Surjectivity
  • In mathematics it is onto functions, epimorphisms in category theory, the surjections counted in combinatorial enumeration (surjections from an n-set onto a k-set), the requirement that a quotient map hit every class, and the covering maps of topology.

    This sourceEpimorphisms as the categorical generalization of surjections; quotient maps and covering maps.

Domain-specific

• Category of representations
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Dual (category theory)
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Free category
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Interchange law
  • … identity and a convenient observable transfers from A rectangle of four natural transformations yields one natural transformation regardless of whether rows or columns are pasted first. to A proof types every source and target and distinguishes strict equality from coherent isomorphism in weak structures..
• Opposite category
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Verification

Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.

Does it back the claim? Read against the text for 1 of 9 citations: 1 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.

Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.

See how references were verified.

Links previously used in the corpus

Before the registry existed this work was also linked 2 other ways.

• https://doi.org/10.1007/978-1-4612-9839-7 ×1
• https://link.springer.com/book/10.1007/978-1-4612-9839-7 ×1

Registry ID ref:3d68e296c0ca · see in the full table