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.
- Thus, ultimately, since categories are interpretations of the axiomatic theories of abstract category
(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
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 0.1
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:
 |
(0.1) |
Remark 0.1
The equivalence classes of such functor representations (defined as natural equivalences) directly 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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