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categorical algebra
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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 structures, 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 categories, functors and natural transformations.
- 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, 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.''
An algebraic representation is generally defined as a morphism $\rho$ from an abstract algebraic structure $\mathcal{A}_S$ to a concrete algebraic structure $A_c$ , a Hilbert space $\mathcal{H}$ , 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 $\mathcal{C}$ 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, $({R}, \phi)$ , which consists of an object $R$ of
$\mathcal{C}$ and a family $\phi$ of equivalences $\phi (C): \Hom_{\mathcal{C}}(R,C) \cong S(C)$ , which is natural in C, with C being any object in $\mathcal{C}$ . When the functor $S$ has such a representation, it is also said to be represented by the object $R$ of $\mathcal{C}$ . For each object $R$ of $\mathbf{C}$ one writes $h_{R}: \mathcal{C} \lra Set$ for the covariant $\Hom$ -functor $h_{R}(C)\cong \Hom_{\mathcal{C}}(R,C)$ . A representation $(R, \phi)$ of ${S}$ is therefore a natural equivalence of functors: \begin{equation} \phi: h_{R} \cong {S}~. \end{equation}
- 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 LMn 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 as quantum LM-algebraic logic, non-commuting graph, non-Abelian structures, topic entry on foundations of mathematics, algebras, algebraic categories and classes of algebras, representable functor, homotopy groupoids and crossed complexes: non-commutative structures in higher dimensional algebra (HDA), index of categories
| 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 |
This object's parent.
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Cross-references: structure, groupoid, algebraic, equivalence classes, natural equivalence, equivalences, object, category of sets, representable, Alexander Grothendieck, key, linear operator, Hilbert space, morphism, ETAS, algebras, ETAC, elementary theory of abstract categories, axiomatic, interpretations, natural transformations, functors, categories, terms, representations, entire, cover, algebraic structures, applications, definitions, field
There are 7 references to this entry.
This is version 67 of categorical algebra, born on 2008-07-17, modified 2009-01-25.
Object id is 10810, canonical name is CategoricalAlgebras.
Accessed 2895 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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