Categories for the Working Mathematician

Mac Lane, S. (1998). Categories for the Working Mathematician. Springer.

Type

Book

Intellectual base

Review or monograph

Year

1998

Volume

5

DOI

10.1007/978-1-4757-4721-8

ISBN

9781441931238

DOI registrant

Springer Nature

Link

https://link.springer.com/book/10.1007/978-1-4757-4721-8

Cited from

mathematics

Cited by

48 citations across 45 artifacts.

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

Primes

• Associativity
  • In category theory, associativity of morphism composition is built into the category axioms; associators in weak higher categories relax strict associativity to associativity-up-to-isomorphism, generalizing the concept via coherence theorems, as Mac Lane (1971) develops in his foundational treatment of monoidal categories.

    This sourceGraduate Texts in Mathematics 5. Springer-Verlag, 1971; 2nd ed. 1998. Develops monoidal categories and the coherence theorem in which strict associativity is relaxed to associativity-up-to-isomorphism (associator, pentagon axiom); Verified.

    Supported in partVerified against the publisher's abstract

    The publisher abstract confirms Mac Lane's book covers coherence theorems for monoidal and higher categories, but never states associators or the pentagon axiom.

    “This second edition includes a number of revisions and additions, including two new chapters on topics of active interest. One is onsymmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence.”

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

• Category
  • An ontology can be read as a category, but they are not the same. Not Hierarchy. A hierarchy is a specific ordered (often tree-like) arrangement; a category is the general objects-arrows-composition structure, of which a partial order is one special case (a category with at most one arrow between objects).

    Supported in partVerified against the publisher's abstract

    Mac Lane's classic treats the concepts of category and its morphisms, but the publisher abstract does not itself state that a partial order is a category with at most one arrow.

    “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

• Commutativity
  • which admit simpler theory than their non-commutative counterparts; the absence of commutativity forces careful sequencing and is itself meaningful — non-commutativity is a feature in contexts where order matters (time-ordered quantum mechanics, sequential message processing); (4) the concept generalizes across domains — mathematics (algebraic axioms; abelian structures; commutative diagrams in category theory, as Mac Lane (1971) systematizes in Categories for the Working Mathematician)

    This sourceGraduate Texts in Mathematics 5. New York: Springer-Verlag, 1971. Standard category-theory text; systematizes the commutative diagram (multiple paths composing to the same morphism) — supports the claim that Mac Lane systematizes commutative diagrams.

    Supported in partVerified against the publisher's abstract

    The publisher's abstract presents Mac Lane's book as category theory's foundational text but never names a commutative diagram, so the claim's specific clause rests on textbook content, not this abstract.

    “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

• Empty Set
  • Without ∅, set-theoretic union is not a monoid; with it, union is monoidal, and monoidal structure is exploited everywhere from databases to functional programming to algebra.

    This sourceDevelops monoid structure — an associative operation with an identity element — making explicit that union is monoidal precisely because ∅ supplies the identity.

    Supported in partVerified against the publisher's abstract

    Mac Lane's book treats categories and monoids and its 2nd edition adds chapters on monoidal categories, but the retrievable abstract says nothing of union, of ∅ as identity, or of database and programming applications.

    “One is onsymmetric monoidal categories and braided monoidal categories and the coherence theorems for them.”

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

• Equivariance
  • The diagram f(g·x) = g·f(x) is the load-bearing relation; everything else is interpretation, a point Mac Lane (1971) makes structurally precise by recasting such commuting squares as naturality conditions.

    This sourceStandard category-theory reference; recasts commuting squares as naturality conditions for natural transformations.

    Supported in partVerified against the publisher's abstract

    The publisher abstract confirms Mac Lane's book treats natural transformation and functor as core concepts, i.e. the frame of the claim, but it does not state that an equivariance square f(g·x) = g·f(x) is recast as a naturality condition.

    “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

• Function (Mapping)
  • … object across mathematics: analysis studies functions of real or complex variables and their limits, derivatives, and integrals; algebra studies functions that preserve structure (homomorphisms, isomorphisms); topology studies continuous functions; measure theory studies measurable functions; and category theory

    This sourceGraduate Texts in Mathematics 5. New York: Springer-Verlag, 1971; 2nd ed., 1998. Standard reference. Precursor: Eilenberg, Samuel, and Saunders Mac Lane. "General Theory of Natural Equivalences." Transactions of the American Mathematical Society 58, no. 2 (September 1945): 231–294, DOI 10.2307/1990284..

• Identity Element
  • A ring exhibits two. Two sides, tested separately. Left and right neutrality are logically independent conditions over the whole carrier, and an operation may satisfy one universally while failing the other everywhere. Uniqueness as consequence, not stipulation. Nothing in the axiom says "exactly one"; the two-sided form yields it, so a structure may host many one-sided identities and never two two-sided ones. Type indexing under partial composition. Where the operation is not total, the role is filled by an indexed family — one neutral element per object, dimension, or schema — rather than a single value.

    This sourceStates the category axioms with an identity arrow assigned to each object and a unit law, and gives Matr_K - objects the positive integers, arrows the rectangular matrices, composition the matrix product - as a category in which each object carries its own identity.

• Relation
  • Category theory

    This sourceGraduate Texts in Mathematics 5. New York: Springer-Verlag, 1971; 2nd ed., 1998. Standard reference. Precursor: Eilenberg, Samuel, and Saunders Mac Lane. "General Theory of Natural Equivalences." Transactions of the American Mathematical Society 58, no. 2 (September 1945): 231–294, DOI 10.2307/1990284..

• Set and Membership
  • The Bourbaki collective's multi-volume Éléments de mathématique systematized this set-theoretic foundation for twentieth-century pure mathematics, though category theory

    This sourceGraduate Texts in Mathematics 5. New York: Springer-Verlag, 1971; 2nd ed., 1998. Standard reference. Precursor: Eilenberg, Samuel, and Saunders Mac Lane. "General Theory of Natural Equivalences." Transactions of the American Mathematical Society 58, no. 2 (September 1945): 231–294, DOI 10.2307/1990284..

• Transformation
  • The concept spans mathematics (linear transformations, affine transformations, group actions, functions, isomorphisms, change of basis), physics (gauge transformations, Lorentz transformations, symmetry transformations and Noether currents), data engineering (extract/transform/load or ETL pipelines), machine learning (feature transformations, normalization, learned representations), chemistry (chemical transformations, retrosynthesis, phase transitions), biology (developmental transformations, metamorphosis, morphogenesis), industry (conversion of raw materials to finished products), language (translation as transformation of linguistic structure), narrative (character arcs, plot transformations), and business (digital transformation, organizational transformation programs), an unusually broad cross-domain pattern that Mac Lane (1971) abstracts as morphisms between objects in his foundational treatment of category theory.

    This sourceGraduate Texts in Mathematics 5. New York: Springer-Verlag, 1971; 2nd ed., 1998. Standard reference. Precursor: Eilenberg, Samuel, and Saunders Mac Lane. "General Theory of Natural Equivalences." Transactions of the American Mathematical Society 58, no. 2 (September 1945): 231–294, DOI 10.2307/1990284..

    Supported in partVerified against the publisher's abstract

    Mac Lane's abstract confirms a foundational treatment of category, functor, natural transformation and morphisms, but not the claim's cross-domain span.

    “Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. 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

Domain-specific

• Associativity Isomorphism
  • Mac Lane's coherence theorem extends the idea: diagrams assembled from associators and unit constraints commute under the monoidal axioms.
• Category of sets
  • The category Set has sets for objects and functions for arrows; identities and associative composition are the ordinary ones, with size conventions used to avoid treating all sets as one set.
• Classification theorem
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Closed monoidal category
  • … transfers from A canonical instance directly demonstrates that the monoidal product, unit, associativity data, side of closure, internal hom, evaluation, and natural adjunction are specified. to An applied instance preserves the same invariant under a changed notation, scale, implementation, or empirical setting..
• Coequalizer
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Cokernel
  • Cokernels are dual to kernels, measure obstruction to surjectivity, and support exact sequences, quotient objects, homology, presentations, and categorical colimits where they exist.
• Concrete category
  • … constitutive identity and a convenient observable transfers from Groups form a concrete category under the forgetful functor sending each group to its underlying set and each homomorphism to its function. to A categorical argument distinguishes a category being concretizable from a chosen concrete presentation..
• Coproduct
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
  • … proposed coproduct object, injection morphisms, arbitrary target and cocone, existence and uniqueness of the mediator, commutative equations and uniqueness up to isomorphism are explicit. to An applied instance preserves the same invariant under changed scale, notation, dataset, jurisdiction, or implementation..
• Diagram (category theory)
  • A diagram of shape J in C is a functor D:J→C; the objects and arrows of J specify the pattern, and D realizes that pattern by objects and morphisms of C.
• Dominant functor
  • … instance directly demonstrates that the source and target categories, functor, quantified target object, source witness, section and retraction arrows and identity composite are explicit. to An applied instance preserves the same invariant under changed scale, notation, jurisdiction, dataset, or implementation..
• Essentially surjective functor
  • Essential surjectivity is weaker than literal objectwise surjectivity, depends on the codomain’s isomorphisms and contributes to equivalence only together with fullness and faithfulness.
• Filtered category
  • A filtered category is a categorical generalization of a directed preorder suitable for filtered colimits.
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Functor
  • The compositional depth of the concept — functors between functor categories, natural transformations as morphisms between functors, adjoint pairs of functors organizing entire mathematical theories — makes it one of the organizing ideas of modern mathematics and theoretical computer science, operating everywhere the notion of "structure-preserving translation between categorically-modeled systems" has grip

    This sourceThe standard reference developing the apparatus this sentence names -- functors and natural transformations (Ch. I), functor categories (Ch. II), and adjoints (Ch. IV); the appraisal of the concept's standing is the article's own.

    Supported in partVerified against the source

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

• Functor Category
  • Once the categorical structure is present, one can form isomorphisms, limits, colimits, adjunctions, and algebraic structures between diagrams as ordinary categorical constructions.
• Image (of a Function)
  • This epi-mono factorization is a basic structural tool across algebra, topology, and category theory

    This sourceThe factorization of an arrow through its image as an epi followed by a monic, used for abelian categories, algebraic quotients and topological spaces.

    SupportedVerified against the source

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

• Injective object
  • For example, the distinction between constitutive identity and a convenient observable transfers from A divisible abelian group is injective in the category of abelian groups. to An injective resolution embeds an object into injectives so derived functors can be computed..
• Isomorphism of categories
  • Categories C and D are isomorphic when functors F:C→D and G:D→C satisfy GF=1_C and FG=1_D exactly.
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Kan extension
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Kernel (category theory)
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Mathematical structure
  • … set equipped with a binary operation, identity, and inverse operation satisfying associativity, identity, and inverse axioms. to The real numbers carry compatible field, order, metric, topological, measurable, and smooth structures that support different classes of morphisms and theorems. demonstrates that continuity.
• Monoid (category theory)
  • … product and unit object, associator and unitors, carrier object, multiplication and unit morphisms, associativity and left and right unit diagrams and commutativity if claimed are explicit. to An applied instance preserves the same invariant under changed scale, notation, jurisdiction, dataset, or implementation..
• Monoidal category action
  • … action bifunctor, action associator and unitor, naturality, invertibility or laxness, pentagon and triangle coherence, morphisms of module categories, and strictness convention are explicit. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, dataset, or implementation..
• Normal Morphism
  • Some references call that a “normal epimorphism”; the present entry uses conormal for the dual to keep the directions visible.
• Pointed set
  • A pointed set is an ordered pair (X,x0) with x0∈X; a pointed map f:(X,x0)→(Y,y0) satisfies f(x0)=y0.
• Pullback (category theory)
  • … that the category and objects, two morphisms and common codomain, proposed object and projections, commutative square, arbitrary cone, unique mediator and uniqueness up to isomorphism are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
• Quotient category
  • Given a category and a congruence on its morphisms, the quotient category has the same objects and hom-sets consisting of equivalence classes of morphisms, with identities and composition induced from the original category.
• Simplex category
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Small set (category theory)
  • … foundation and universe, membership and closure axioms, set or class under discussion, relative size level, universe change and distinction from finite and small-category usage are explicit. to An applied instance preserves the same invariant under changed scale, notation, dataset, jurisdiction, or implementation..
• Subcategory
  • … selected objects and morphisms, source and target inheritance, identity inclusion, composition closure, size convention, and any full, wide, replete, or induced qualifier are explicit. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, dataset, or implementation..
• Subobject
  • A subobject of A is represented by a monomorphism m:S→A, with two representatives identified when an isomorphism between their domains commutes with the embeddings.
• Subquotient
  • A subquotient of an object is a quotient object of one of its subobjects; for groups it has form G′/N with G′≤G and N normal in G′.
• Transport of Structure
  • Mac Lane's algebraic theories and their algebras formalize operations as product-preserving functors.
• Universal property
  • The free group on a set \(S\) is the group \(F(S)\) such that any function from \(S\) to any group \(G\) extends to a unique group homomorphism from \(F(S)\) to \(G\); limits, colimits, adjoints, Stone–Čech compactifications, and representable functors in algebraic geometry all carry universal properties of the same structural type

    This sourceUniversal arrows as the common form of free objects, limits, colimits, adjoints and representable functors. The results that adjoints compose (IV.8) and that limits interchange with limits (IX.2), both derived from universal properties.

    Supported in partVerified against the source

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

Mechanisms

• Commutative Diagram Review
  • It borrows its rigor from the commutative diagram of category theory, where a diagram "commutes" exactly when all directed paths between two objects compose to the same map.

    This sourceDefines commutativity as equality of the composites produced by directed paths with common endpoints.

• Source/Target Type Check
  • It is the front-line gate every composition depends on.

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 9 of 48 citations: 1 supported, 8 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 3 other ways.

• https://doi.org/10.1007/978-1-4757-4721-8 ×10
• https://doi.org/10.1007/978-1-4612-9839-7 ×1
• https://math.mit.edu/~hrm/palestine/maclane-categories.pdf ×1

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