# 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.

## 0.3 Basic definitions

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 (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 $\mathcal{C}$ 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 $\phi(C):{\rm Hom}_{\mathcal{C}}(R,C)\cong S(C)$, 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 $h_{R}:\mathcal{C}{\longrightarrow}Set$ for the covariant ${\rm Hom}$–functor $h_{R}(C)\cong{\rm Hom}_{\mathcal{C}}(R,C)$. A representation $(R,\phi)$ of ${S}$ is therefore a natural equivalence of functors:

 $\phi:h_{R}\cong{S}~{}.$ (0.1)

## 0.4 Note:

 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