Categories for the Working Mathematician

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

Type

Book

Intellectual base

Review or monograph

Year

1998

DOI

10.1007/978-1-4757-4721-8

ISBN

9781441931238

Link

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

Cited from

computer_science, 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)

• 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).
• 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.

• 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.

• 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.

• 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.

• 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

• 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.

• 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.

• 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.

• 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

• 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

• 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.

• 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

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.

• Functor Category
  • Once the categorical structure is present, one can form isomorphisms, limits, colimits, adjunctions, and algebraic structures between diagrams as ordinary categorical constructions.
• 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..
• 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.

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.

Verification

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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 ×2
• https://link.springer.com/book/10.1007/978-1-4612-9839-7 ×1

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