categorical algebra

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 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

.

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

generalization

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 set

.

0.2

Extensions of categorical algebra

•

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

.”

0.3

Basic definitions

An

algebraic representation

is generally defined as a

morphism

ρ

from an abstract algebraic structure

A

S

to a concrete algebraic structure

A

c

, a Hilbert space

ℋ

, or a family of linear operator spaces.

The key notion of

representable functor

(

http://planetmath.org/RepresentableFunctor

) 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

S

⁢

e

⁢

t

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

)

:

Hom

𝒞

⁢

(

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

h

R

:

𝒞

⟶

S

⁢

e

⁢

t

for the covariant

Hom

–functor

h

R

⁢

(

C

)

≅

Hom

𝒞

⁢

(

R

,

C

)

. A

representation

(

R

,

ϕ

)

of

S

is therefore

a

natural equivalence

of functors

:

ϕ

:

h

R

≅

S

.

(0.1)

Remark 0.1

.

The

equivalence classes

of such functor representations (defined as natural equivalences) determine directly an algebraic (

groupoid

)

structure

.

0.4

Note:

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

0.5

Application: Quantum Categories

References

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.

3

http://en.wikipedia.org/wiki/Categorical_algebra

Categorical

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 categories

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

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