categorical algebra

0.1 Introduction: An Outline of Categorical Algebra

This topic entry provides an outline of an important mathematical field called categorical algebra; although specific definitions are in use for various applications of categorical algebra to specific algebraic structuresPlanetmathPlanetmath, they do not cover the entire field. In the most general sense, categorical algebras– as introduced by Mac Lane in 1965 – can be described as the study of representations of algebraic structures, either concrete or abstract, in terms of categoriesMathworldPlanetmath, functorsMathworldPlanetmath and natural transformations.

In a narrow sense, a categorical algebra is an associative algebra, defined for any locally finite category and a commutative ring with unity. This notion may be considered as a generalizationPlanetmathPlanetmath of both the concept of group algebra and that of an incidence algebra, much as the concept of category generalizes the notions of group and partially ordered setMathworldPlanetmath.

0.2 Extensions of categorical algebra

  • Thus, ultimately, since categories are interpretationsMathworldPlanetmathPlanetmath of the axiomatic elementary theory of abstract categories (ETAC), so are categorical algebras.

    The general definition of representation introduced above can be still further extended by introducing supercategorical algebrasMathworldPlanetmath as interpretations of ETAS, as explained next.

  • Mac Lane (1976) wrote in his Bull. AMS review cited here as a verbatim quotation:

    “On some occasions I have been tempted to try to define what algebra is, can, or should be - most recently in concluding a survey [72] on Recent advances in algebra. But no such formal definitions hold valid for long, since algebra and its various subfields steadily change under the influence of ideas and problems coming not just from logic and geometry, but from analysis, other parts of mathematics, and extra mathematical sources. The progress of mathematics does indeed depend on many interlocking, unexpected and multiform developments.”

0.3 Basic definitions

An algebraic representation is generally defined as a morphismMathworldPlanetmath ρ from an abstract algebraic structure AS to a concrete algebraic structure Ac, a Hilbert space , or a family of linear operator spaces.

The key notion of representable functor ( was first reported by Alexander Grothendieck in 1960.

Definition 0.1.

Thus, when the latter concept is extended to categorical algebra, one has a representable functor S from an arbitrary category 𝒞 to the category of sets Set if S admits a functor representation defined as follows. A functor representation of S is defined as a pair, (R,ϕ), which consists of an object R of 𝒞 and a family ϕ of equivalences ϕ(C):Hom𝒞(R,C)S(C), which is natural in C, with C being any object in 𝒞. When the functor S has such a representation, it is also said to be represented by the object R of 𝒞. For each object R of 𝐂 one writes hR:𝒞Set for the covariant Hom–functor hR(C)Hom𝒞(R,C). A representation (R,ϕ) of S is therefore a natural equivalence of functors:

ϕ:hRS. (0.1)
Remark 0.1.

The equivalence classesMathworldPlanetmathPlanetmath of such functor representations (defined as natural equivalences) determine directly an algebraic (groupoidPlanetmathPlanetmathPlanetmathPlanetmath) structure.

0.4 Note:

See also in Expositions the entry about abstract and concrete algebras.

0.5 Application: Quantum Categories


  • 1 Saunders Mac Lane: Categorical algebra., Bull. AMS, 71 (1965), 40-106., Zbl 0161.01601, MR 0171826,
  • 2 Saunders Mac Lane: TopologyMathworldPlanetmath and Logic as a Source of Algebras., Bull. AMS, 82, Number 1, 1-36, January 1, 1976.
  • 3 algebra- basic definitions
Title categorical algebra
Canonical name CategoricalAlgebra
Date of creation 2013-03-22 18:13:27
Last modified on 2013-03-22 18:13:27
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 79
Author bci1 (20947)
Entry type Topic
Classification msc 08A99
Classification msc 08A05
Classification msc 08A70
Synonym algebraic categoriesPlanetmathPlanetmathPlanetmath
Related topic AlgebraicCategoryOfLMnLogicAlgebras
Related topic NonAbelianStructures
Related topic AbelianCategory
Related topic AxiomsForAnAbelianCategory
Related topic GeneralizedVanKampenTheoremsHigherDimensional
Related topic AxiomaticTheoryOfSupercategories
Related topic CategoricalOntology
Related topic NonCommutingGraphOfAGroup
Related topic NonAbelianStructures
Defines algebraic representation
Defines functor representation
Defines representable functor
Defines category of algebraic structures
Defines category of logic algebras