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categorical algebra
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(Topic)
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The following is a generic definition or description of the mathematical field called `Categorical Algebra'; more specific definitions are possible and are being used in various applications of algebraic structures.
In the most general sense, categorical algebras (e.g, as introduced by Mac Lane, 1965) are being generically defined as representations of algebraic structures in terms of categories, functors and natural transformations.
Remarks
Thus, ultimately, since categories are interpretations 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 algebras as interpretations of ETAS.
Thus, 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.”
Definition 0.1 An algebraic representation is generally defined as a morphism  from an abstract algebraic structure
 to a concrete algebraic structure  , 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 0.2 Thus, when the latter concept is extended to Categorical Algebra, one has a representable functor  from an arbitrary category
 to the category of sets  if  admits a functor representation defined as follows. A functor representation of  is defined as a pair,
 , which consists of an object  of
 and a family  of equivalences
 , which is natural in C, with C being any object in
 . When the functor  has such a representation, it is also said to be represented by the object  of
 . For each object  of
 one writes
 for the covariant  -functor
 . A representation  of  is therefore a natural equivalence of functors:
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(0.1) |
The equivalence classes of such functor representations (defined as natural equivalences) obviously determine an algebraic (groupoid) structure.
- 1
- Saunders Mac Lane: Categorical algebra., Bull. AMS, 71 (1965), 40-106., Zbl 0161.01601, MR 0171826,
- 2
- Saunders Mac Lane: Topology and Logic as a Source of Algebras., Bull. AMS, 82, Number 1, 1-36, January 1, 1976.
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"categorical algebra" is owned by bci1. [ full author list (2) ]
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See Also: algebraic category of  -logic algebras, non-Abelian structures, abelian category, supplemental axioms for an Abelian category, higher dimensional generalized Van Kampen theorems (HD-VKT), axiomatic theory of supercategories and metacategories, categorical quantum logics: quantum LM-algebraic logic, non-commuting graph, non-Abelian structures, topic entry on foundations of mathematics, algebra classification, classes of algebras, representable functor,  -supercategories, homotopy groupoids and crossed complexes: non-commutative structures in higher dimensional algebra (HDA)
| Other names: |
algebraic categories |
| Also defines: |
algebraic representation, functor representation, representable functor, category of algebraic structures, category of logic algebras |
| Keywords: |
representations, categorical algebra, algebra or algebraic representations, representable functors, algebraic categories, categories of algebraic structures, categories of logic algebras, universal-algebras, operator algebras |
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Cross-references: structure, groupoid, algebraic, equivalence classes, natural equivalence, equivalences, object, category of sets, representable, Alexander Grothendieck, linear operator, Hilbert space, morphism, ETAS, supercategorical algebras, ETAC, theory, axiomatic, interpretations, natural transformations, functors, categories, terms, representations, generically, algebraic structures, applications, definitions, field, generic
There are 16 references to this entry.
This is version 54 of categorical algebra, born on 2008-07-17, modified 2008-10-11.
Object id is 10810, canonical name is CategoricalAlgebras.
Accessed 1179 times total.
Classification:
| AMS MSC: | 08A70 (General algebraic systems :: Algebraic structures :: Applications of universal algebra in computer science) | | | 08A05 (General algebraic systems :: Algebraic structures :: Structure theory) | | | 08A99 (General algebraic systems :: Algebraic structures :: Miscellaneous) |
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Pending Errata and Addenda
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